1.1.1 |
x1+4x2 | = | 5 |
4x1+5x2 | = | −13 |
1.1.1 |
x1+5x2 | = | 9 |
4x1+5x2 | = | −9 |
1.1.2 |
5x1+10x2 | = | 20 |
4x1+7x2 | = | 18 |
1.1.2 |
6x1+12x2 | = | 18 |
4x1+7x2 | = | 15 |
1.1.3 |
Find the point x1,x2 that lies on the line x1+2x2=5 and on the line x1−x2=−1. See the figure. | x1 x2 |
1.1.3 |
Find the point x1,x2 that lies on the line x1+2x2=10 and on the line x1−x2=1. See the figure. | x1 x2 |
1.1.13 |
x1 | − | 3x3 | = | 7 | |||
4x1 | + | 4x2 | + | x3 | = | 23 | |
2x2 | + | 3x3 | = | 1 |
1.1.13 |
x1 | − | 6x3 | = | 22 | |||
2x1 | + | 4x2 | + | x3 | = | 21 | |
2x2 | + | 3x3 | = | −1 |
1.1.26 |
| 6 | 4 | h |
| −3 | −2 | 5 |
1.1.26 |
| 6 | 4 | h |
| −3 | −2 | 3 |
1.1.28 |
1.1.31 |
1.1.31 |
1.2.2 |
a.
| b.
| c.
|
1.2.2 |
a.
| b.
| c.
|
1.2.2 |
a.
| b.
| c.
|
1.2.2 |
a.
| b.
| c.
|
1.2.2 |
a.
| b.
| c.
|
1.2.5 |
1.2.5 |
1.2.6 |
1.2.6 |
1.2.7 |
| 1 | 3 | 4 | 8 | ||
| 2 | 6 | 5 | 7 |
| x1=−4−3x2 |
| x2 is free |
| x3=3 |
| x1= |
| x2 is free |
| x3 is free |
| x1= |
| x2= |
| x3= |
1.2.7 |
| 1 | 4 | 3 | 3 | ||
| 2 | 8 | 1 | −4 |
| x1= |
| x2 is free |
| x3 is free |
| x1= |
| x2= |
| x3= |
| x1=−3−4x2 |
| x2 is free |
| x3=2 |
1.2.8 |
| 1 | 4 | 0 | 9 | ||
| 2 | 7 | 0 | 15 |
| x1=−3 |
| x2=3 |
| x3 is free |
| x1= |
| x2 is free |
| x3 is free |
| x1= |
| x2= |
| x3= |
1.2.8 |
| 1 | 1 | 0 | −4 | ||
| 3 | 2 | 0 | −15 |
| x1=−7 |
| x2=3 |
| x3 is free |
| x1= |
| x2= |
| x3= |
| x1= |
| x2 is free |
| x3 is free |
1.2.9 |
| 0 | 1 | −4 | 6 | ||
| 1 | −4 | 14 | −21 |
| x1= |
| x2 is free |
| x3 is free |
| x1= |
| x2= |
| x3= |
| x1=3+2x3 |
| x2=6+4x3 |
| x3 is free |
1.2.9 |
| 0 | 1 | −2 | 3 | ||
| 1 | −3 | 1 | −5 |
| x1=4+5x3 |
| x2=3+2x3 |
| x3 is free |
| x1= |
| x2= |
| x3= |
| x1= |
| x2 is free |
| x3 is free |
1.2.11 |
| 3 | −2 | 7 | 0 |
| 12 | −8 | 28 | 0 |
| 9 | −6 | 21 | 0 |
| x1=23x2−73x3 |
| x2 is free |
| x3 is free |
| x1=−3x2 |
| x2=2x3 |
| x3 is free |
| x1=3 |
| x2=−2 |
| x3=7 |
1.2.11 |
| 5 | −3 | 7 | 0 |
| 10 | −6 | 14 | 0 |
| 15 | −9 | 21 | 0 |
| x1=35x2−75x3 |
| x2 is free |
| x3 is free |
| x1=−5x2 |
| x2=3x3 |
| x3 is free |
| x1=5 |
| x2=−3 |
| x3=7 |
1.2.11 |
| 2 | −3 | 9 | 0 |
| 4 | −6 | 18 | 0 |
| 6 | −9 | 27 | 0 |
| x1=32x2−92x3 |
| x2 is free |
| x3 is free |
| x1=−2x2 |
| x2=3x3 |
| x3 is free |
| x1=2 |
| x2=−3 |
| x3=9 |
1.2.11 |
| 3 | −2 | 4 | 0 |
| 9 | −6 | 12 | 0 |
| 6 | −4 | 8 | 0 |
| x1=23x2−43x3 |
| x2 is free |
| x3 is free |
| x1=−3x2 |
| x2=2x3 |
| x3 is free |
| x1=3 |
| x2=−2 |
| x3=4 |
1.2.13 |
| 1 | −2 | 0 | −1 | 0 | −8 |
| 0 | 1 | 0 | 0 | −4 | 7 |
| 0 | 0 | 0 | 1 | 4 | 5 |
| 0 | 0 | 0 | 0 | 0 | 0 |
| x1= |
| x2 is free |
| x3= |
| x4 is free |
| x5 is free |
| x1= |
| x2= |
| x3 is free |
| x4= |
| x5= |
| x1=11+4x5 |
| x2=7+4x5 |
| x3 is free |
| x4=5−4x5 |
| x5 is free |
1.2.13 |
| 1 | −5 | 0 | −1 | 0 | −4 |
| 0 | 1 | 0 | 0 | −3 | 1 |
| 0 | 0 | 0 | 1 | 3 | 2 |
| 0 | 0 | 0 | 0 | 0 | 0 |
| x1=3+12x5 |
| x2=1+3x5 |
| x3 is free |
| x4=2−3x5 |
| x5 is free |
| x1= |
| x2= |
| x3 is free |
| x4= |
| x5= |
| x1= |
| x2 is free |
| x3= |
| x4 is free |
| x5 is free |
1.2.13 |
| 1 | −2 | 0 | −1 | 0 | −9 |
| 0 | 1 | 0 | 0 | −3 | 5 |
| 0 | 0 | 0 | 1 | 8 | 9 |
| 0 | 0 | 0 | 0 | 0 | 0 |
| x1=10−2x5 |
| x2=5+3x5 |
| x3 is free |
| x4=9−8x5 |
| x5 is free |
| x1= |
| x2= |
| x3 is free |
| x4= |
| x5= |
| x1= |
| x2 is free |
| x3= |
| x4 is free |
| x5 is free |
1.2.13 |
| 1 | −7 | 0 | −1 | 0 | −3 |
| 0 | 1 | 0 | 0 | −2 | 3 |
| 0 | 0 | 0 | 1 | 7 | 2 |
| 0 | 0 | 0 | 0 | 0 | 0 |
| x1= |
| x2 is free |
| x3= |
| x4 is free |
| x5 is free |
| x1= |
| x2= |
| x3 is free |
| x4= |
| x5= |
| x1=20+7x5 |
| x2=3+2x5 |
| x3 is free |
| x4=2−7x5 |
| x5 is free |
1.2.13 |
| 1 | −7 | 0 | −1 | 0 | −2 |
| 0 | 1 | 0 | 0 | −3 | 1 |
| 0 | 0 | 0 | 1 | 4 | 5 |
| 0 | 0 | 0 | 0 | 0 | 0 |
| x1= |
| x2= |
| x3 is free |
| x4= |
| x5= |
| x1=10+17x5 |
| x2=1+3x5 |
| x3 is free |
| x4=5−4x5 |
| x5 is free |
| x1= |
| x2 is free |
| x3= |
| x4 is free |
| x5 is free |
1.2.13 |
| 1 | −4 | 0 | −1 | 0 | −6 |
| 0 | 1 | 0 | 0 | −2 | 1 |
| 0 | 0 | 0 | 1 | 6 | 7 |
| 0 | 0 | 0 | 0 | 0 | 0 |
| x1= |
| x2= |
| x3 is free |
| x4= |
| x5= |
| x1=5+2x5 |
| x2=1+2x5 |
| x3 is free |
| x4=7−6x5 |
| x5 is free |
| x1= |
| x2 is free |
| x3= |
| x4 is free |
| x5 is free |
1.2.13 |
| 1 | −3 | 0 | −1 | 0 | −6 |
| 0 | 1 | 0 | 0 | −9 | 7 |
| 0 | 0 | 0 | 1 | 8 | 5 |
| 0 | 0 | 0 | 0 | 0 | 0 |
| x1= |
| x2= |
| x3 is free |
| x4= |
| x5= |
| x1=20+19x5 |
| x2=7+9x5 |
| x3 is free |
| x4=5−8x5 |
| x5 is free |
| x1= |
| x2 is free |
| x3= |
| x4 is free |
| x5 is free |
1.2.14 |
| 1 | 0 | −5 | 0 | −6 | 4 |
| 0 | 1 | 8 | −1 | 0 | 3 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 |
| x1= |
| x2= |
| x3= |
| x4= |
| x5 is free |
| x1= |
| x2= |
| x3 is free |
| x4= |
| x5= |
| x1=4+5x3 |
| x2=3+x4−8x3 |
| x3 is free |
| x4 is free |
| x5=0 |
1.2.14 |
| 1 | 0 | −5 | 0 | −4 | 8 |
| 0 | 1 | 6 | −1 | 0 | 7 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 |
| x1=8+5x3 |
| x2=7+x4−6x3 |
| x3 is free |
| x4 is free |
| x5=0 |
| x1= |
| x2= |
| x3= |
| x4= |
| x5 is free |
| x1= |
| x2= |
| x3 is free |
| x4= |
| x5= |
1.2.14 |
| 1 | 0 | −5 | 0 | −4 | 7 |
| 0 | 1 | 3 | −1 | 0 | 2 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 |
| x1= |
| x2= |
| x3 is free |
| x4= |
| x5= |
| x1=7+5x3 |
| x2=2+x4−3x3 |
| x3 is free |
| x4 is free |
| x5=0 |
| x1= |
| x2= |
| x3= |
| x4= |
| x5 is free |
1.2.14 |
| 1 | 0 | −6 | 0 | −7 | 4 |
| 0 | 1 | 3 | −1 | 0 | 5 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 |
| x1=4+6x3 |
| x2=5+x4−3x3 |
| x3 is free |
| x4 is free |
| x5=0 |
| x1= |
| x2= |
| x3= |
| x4= |
| x5 is free |
| x1= |
| x2= |
| x3 is free |
| x4= |
| x5= |
1.2.22 |
| 1 | h | 2 |
| −4 | 20 | −10 |
1.2.22 |
| 1 | h | 3 |
| 2 | 10 | 3 |
1.2.25 |
1.2.25 |
1.2.26 |
1.2.26 |
1.2.26 |
1.2.29 |
1.2.29 |
1.2.29 |
1.2.29 |
1.2.31 |
1.2.31 |
1.2.34 |
1.2.34 |
1.2.35 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 |
1.2.35 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
1.2.36 |
| 0 | ⋯ | 0 | b |
| 0 | 0 | 0 | 0 | b |
1.2.36 |
| 0 | ⋯ | 0 | b |
| 0 | 0 | 0 | 0 | b |
1.2.37 |
| 0 | ⋯ | 0 | b |
1.2.37 |
| 0 | ⋯ | 0 | b |
1.2.45 |
a0 | + | a1(1) | + | a2(1)2 | = | 12 |
a0 | + | a1(2) | + | a2(2)2 | = | 15 |
a0 | + | a1(3) | + | a2(3)2 | = | 16 |
1.2.45 |
a0 | + | a1(1) | + | a2(1)2 | = | 11 |
a0 | + | a1(2) | + | a2(2)2 | = | 19 |
a0 | + | a1(3) | + | a2(3)2 | = | 21 |
1.2.45 |
a0 | + | a1(1) | + | a2(1)2 | = | 14 |
a0 | + | a1(2) | + | a2(2)2 | = | 18 |
a0 | + | a1(3) | + | a2(3)2 | = | 20 |
1.3.5 |
| 2 | ||
| −4 | ||
| 6 |
| 6 | ||
| 0 | ||
| −8 |
| 4 | ||
| −5 | ||
| 7 |
2x1 | + | 6x2 | = | 4 |
−4x1 | + | x2 | = | −5 |
6x1 | − | 8x2 | = | 7 |
2x1 | + | 6x2 | = | 7 |
−4x1 | = | −5 | ||
6x1 | − | 8x2 | = | 4 |
2x1 | + | 6x2 | = | 4 |
−4x1 | = | −5 | ||
6x1 | − | 8x2 | = | 7 |
2x1 | + | 6x2 | = | −5 |
−4x1 | = | 4 | ||
6x1 | − | 8x2 | = | 7 |
1.3.5 |
| 4 | ||
| −4 | ||
| 7 |
| 6 | ||
| 0 | ||
| −8 |
| 2 | ||
| −2 | ||
| 9 |
4x1 | + | 6x2 | = | 2 |
−4x1 | + | x2 | = | −2 |
7x1 | − | 8x2 | = | 9 |
4x1 | + | 6x2 | = | 2 |
−4x1 | = | −2 | ||
7x1 | − | 8x2 | = | 9 |
4x1 | + | 6x2 | = | −2 |
−4x1 | = | 2 | ||
7x1 | − | 8x2 | = | 9 |
4x1 | + | 6x2 | = | 9 |
−4x1 | = | −2 | ||
7x1 | − | 8x2 | = | 2 |
1.3.6 |
| 5 |
| −2 |
| 6 |
| 4 |
| −5 |
| 1 |
| 0 |
| 0 |
5x1 | + | 6x2 | + | 5x3 | = | 0 |
2x1 | + | 4x2 | + | x3 | = | 0 |
5x1 | + | 6x2 | − | 5x3 | = | 0 |
−2x1 | + | 4x2 | = | 0 |
5x1 | + | 6x2 | + | 5x3 | = | 0 |
2x1 | + | 4x2 | = | 0 |
5x1 | + | 6x2 | − | 5x3 | = | 0 |
−2x1 | + | 4x2 | + | x3 | = | 0 |
1.3.6 |
| 4 |
| −3 |
| 7 |
| 4 |
| −2 |
| 1 |
| 0 |
| 0 |
4x1 | + | 7x2 | − | 2x3 | = | 0 |
−3x1 | + | 4x2 | + | x3 | = | 0 |
4x1 | + | 7x2 | + | 2x3 | = | 0 |
3x1 | + | 4x2 | + | x3 | = | 0 |
4x1 | + | 7x2 | + | 2x3 | = | 0 |
3x1 | + | 4x2 | = | 0 |
4x1 | + | 7x2 | − | 2x3 | = | 0 |
−3x1 | + | 4x2 | = | 0 |
1.3.9 |
x2 | + | 3x3 | = | 0 | ||
5x1 | + | 9x2 | − | x3 | = | 0 |
−x1 | + | 4x2 | − | 6x3 | = | 0 |
| 0 |
| 5 |
| −1 |
| 1 |
| 9 |
| 4 |
| 0 |
| 0 |
| 0 |
| 3 |
| −1 |
| −6 |
| 1 |
| 9 |
| 4 |
| 0 |
| 5 |
| −1 |
| 3 |
| −1 |
| −6 |
| 0 |
| 0 |
| 0 |
| 0 |
| 5 |
| −1 |
| 3 |
| −1 |
| −6 |
| 1 |
| 9 |
| 4 |
| 0 |
| 0 |
| 0 |
| 0 |
| 5 |
| −1 |
| 1 |
| 9 |
| 4 |
| 3 |
| −1 |
| −6 |
| 0 |
| 0 |
| 0 |
1.3.9 |
x2 | + | 2x3 | = | 0 | ||
4x1 | + | 6x2 | − | x3 | = | 0 |
−x1 | + | 3x2 | − | 8x3 | = | 0 |
| 0 |
| 4 |
| −1 |
| 2 |
| −1 |
| −8 |
| 1 |
| 6 |
| 3 |
| 0 |
| 0 |
| 0 |
| 0 |
| 4 |
| −1 |
| 1 |
| 6 |
| 3 |
| 2 |
| −1 |
| −8 |
| 0 |
| 0 |
| 0 |
| 0 |
| 4 |
| −1 |
| 1 |
| 6 |
| 3 |
| 0 |
| 0 |
| 0 |
| 2 |
| −1 |
| −8 |
| 1 |
| 6 |
| 3 |
| 0 |
| 4 |
| −1 |
| 2 |
| −1 |
| −8 |
| 0 |
| 0 |
| 0 |
1.3.9 |
x2 | + | 2x3 | = | 0 | ||
5x1 | + | 8x2 | − | x3 | = | 0 |
−x1 | + | 3x2 | − | 8x3 | = | 0 |
| 1 |
| 8 |
| 3 |
| 0 |
| 5 |
| −1 |
| 2 |
| −1 |
| −8 |
| 0 |
| 0 |
| 0 |
| 0 |
| 5 |
| −1 |
| 1 |
| 8 |
| 3 |
| 0 |
| 0 |
| 0 |
| 2 |
| −1 |
| −8 |
| 0 |
| 5 |
| −1 |
| 2 |
| −1 |
| −8 |
| 1 |
| 8 |
| 3 |
| 0 |
| 0 |
| 0 |
| 0 |
| 5 |
| −1 |
| 1 |
| 8 |
| 3 |
| 2 |
| −1 |
| −8 |
| 0 |
| 0 |
| 0 |
1.3.9 |
x2 | + | 5x3 | = | 0 | ||
5x1 | + | 6x2 | − | x3 | = | 0 |
−x1 | + | 3x2 | − | 8x3 | = | 0 |
| 0 |
| 5 |
| −1 |
| 5 |
| −1 |
| −8 |
| 1 |
| 6 |
| 3 |
| 0 |
| 0 |
| 0 |
| 0 |
| 5 |
| −1 |
| 1 |
| 6 |
| 3 |
| 5 |
| −1 |
| −8 |
| 0 |
| 0 |
| 0 |
| 0 |
| 5 |
| −1 |
| 1 |
| 6 |
| 3 |
| 0 |
| 0 |
| 0 |
| 5 |
| −1 |
| −8 |
| 1 |
| 6 |
| 3 |
| 0 |
| 5 |
| −1 |
| 5 |
| −1 |
| −8 |
| 0 |
| 0 |
| 0 |
1.3.11 |
| 1 |
| −3 |
| 0 |
| 0 |
| 1 |
| 4 |
| 5 |
| −6 |
| 36 |
| 2 |
| −1 |
| 20 |
1.3.11 |
| 1 |
| −4 |
| 0 |
| 0 |
| 1 |
| 3 |
| 5 |
| −6 |
| 42 |
| 3 |
| −2 |
| 30 |
1.3.11 |
| 1 |
| −3 |
| 0 |
| 0 |
| 1 |
| 2 |
| 5 |
| −5 |
| 20 |
| 4 |
| −1 |
| 22 |
1.3.11 |
| 1 |
| −2 |
| 0 |
| 0 |
| 1 |
| 4 |
| 6 |
| −5 |
| 28 |
| 3 |
| −2 |
| 16 |
1.3.13 |
| 1 | −3 | 4 |
| 0 | 2 | 5 |
| −3 | 9 | −12 |
| 4 |
| −5 |
| −4 |
1.3.15 |
| 6 | ||
| 1 | ||
| −7 |
| −5 | ||
| 3 | ||
| 0 |
| 0 | ||
| 0 | ||
| 0 |
| 6 | ||
| 1 | ||
| −7 |
| −5 | ||
| 3 | ||
| 0 |
| 1 | ||
| 4 | ||
| −7 |
| 11 | ||
| −2 | ||
| −7 |
1.3.15 |
| 6 | ||
| 2 | ||
| −7 |
| −4 | ||
| 4 | ||
| 0 |
| 0 | ||
| 0 | ||
| 0 |
| 6 | ||
| 2 | ||
| −7 |
| −4 | ||
| 4 | ||
| 0 |
| 2 | ||
| 6 | ||
| −7 |
| 10 | ||
| −2 | ||
| −7 |
1.3.27 |
1.3.32 |
| a1 | a2 | a3 | b |
| a1 | a2 | a3 | b |
| a1 | a2 | a3 | b |
| a1 | a2 | a3 | b |
1.3.32 |
| a1 | a2 | a3 | b |
| a1 | a2 | a3 | b |
| a1 | a2 | a3 | b |
| a1 | a2 | a3 | b |
1.3.32 |
| a1 | a2 | a3 | b |
| a1 | a2 | a3 | b |
| a1 | a2 | a3 | b |
| a1 | a2 | a3 | b |
1.3.32 |
| a1 | a2 | a3 | b |
| a1 | a2 | a3 | b |
| a1 | a2 | a3 | b |
| a1 | a2 | a3 | b |
1.3.33 |
| 1 | 0 | −6 | ||
| 0 | 3 | −5 | ||
| −6 | 9 | 5 |
| 4 | ||
| −4 | ||
| −20 |
1.3.33 |
| 1 | 0 | −5 | ||
| 0 | 2 | −3 | ||
| −4 | 4 | 5 |
| 7 | ||
| 4 | ||
| −2 |
1.4.7 |
| −9 |
| 8 |
| 4 |
| −6 |
| −1 |
| 8 |
| −9 |
| 4 |
| 6 |
| −8 |
| −7 |
| −4 |
| 5 |
| 3 |
| 1 |
| 7 |
| −9 | −1 | 6 | ||
| 8 | 8 | −8 | ||
| 4 | −9 | −7 | ||
| −6 | 4 | −4 |
| x1 | ||
| x2 | ||
| x3 |
| 5 | ||
| 3 | ||
| 1 | ||
| 7 |
1.4.7 |
| 2 |
| 5 |
| −6 |
| −2 |
| 0 |
| −4 |
| −4 |
| −7 |
| −2 |
| 7 |
| 2 |
| −2 |
| 9 |
| 8 |
| 1 |
| 3 |
| 2 | 0 | −2 | ||
| 5 | −4 | 7 | ||
| −6 | −4 | 2 | ||
| −2 | −7 | −2 |
| x1 | ||
| x2 | ||
| x3 |
| 9 | ||
| 8 | ||
| 1 | ||
| 3 |
1.4.8 |
| −1 | ||
| 1 |
| −5 | ||
| −5 |
| −4 | ||
| 0 |
| 5 | ||
| 5 |
| 3 | ||
| −1 |
| −1 | −5 | −4 | 5 | ||
| 1 | −5 | 0 | 5 |
| z1 | ||
| z2 | ||
| z3 | ||
| z4 |
| 3 | ||
| −1 |
1.4.8 |
| −4 | ||
| 3 |
| −2 | ||
| −1 |
| −3 | ||
| 5 |
| 2 | ||
| 5 |
| 3 | ||
| 18 |
| −4 | −2 | −3 | 2 | ||
| 3 | −1 | 5 | 5 |
| z1 | ||
| z2 | ||
| z3 | ||
| z4 |
| 3 | ||
| 18 |
1.4.9 |
8x1 | + | x2 | − | 3x3 | = | 8 |
6x2 | + | 10x3 | = | 0 |
| x1 | ||
| x2 | ||
| x3 |
| 8 | ||
| 0 |
| 1 | ||
| 6 |
| −3 | ||
| 10 |
| 8 | ||
| 0 |
| x1 | x2 | x3 |
| x1 | x2 | x3 |
| 8 | 1 | −3 | ||
| 0 | 6 | 10 |
| x1 | ||
| x2 | ||
| x3 |
| 8 | ||
| 0 |
1.4.10 |
9x1 | − | x2 | = | 5 |
4x1 | + | 5x2 | = | 4 |
10x1 | − | x2 | = | 1 |
| x1 | ||
| x2 |
| x1 | x2 |
| 9 | ||
| 4 | ||
| 10 |
| −1 | ||
| 5 | ||
| −1 |
| 5 | ||
| 4 | ||
| 1 |
| x1 | x2 |
| 9 | −1 | ||
| 4 | 5 | ||
| 10 | −1 |
| x1 | ||
| x2 |
| 5 | ||
| 4 | ||
| 1 |
1.4.10 |
2x1 | − | x2 | = | 10 |
5x1 | + | 6x2 | = | 4 |
9x1 | − | x2 | = | 1 |
| x1 | ||
| x2 |
| 2 | ||
| 5 | ||
| 9 |
| −1 | ||
| 6 | ||
| −1 |
| 10 | ||
| 4 | ||
| 1 |
| x1 | x2 |
| 2 | −1 | ||
| 5 | 6 | ||
| 9 | −1 |
| x1 | ||
| x2 |
| 10 | ||
| 4 | ||
| 1 |
| x1 | x2 |
1.4.11 |
Given A and b to the right, write the augmented matrix for the linear system that corresponds to the matrix equation Ax=b. Then solve the system and write the solution as a vector. | A=
b=
|
| 1 | 2 | −3 | 4 | ||
| −2 | −5 | 2 | −13 | ||
| −3 | −2 | 4 | 8 |
| x1 | ||
| x2 | ||
| x3 |
| −6 | ||
| 5 | ||
| 0 |
1.4.11 |
Given A and b to the right, write the augmented matrix for the linear system that corresponds to the matrix equation Ax=b. Then solve the system and write the solution as a vector. | A=
b=
|
| 1 | 4 | −3 | 2 | ||
| −2 | −2 | 0 | 14 | ||
| 3 | 2 | 4 | −24 |
| x1 | ||
| x2 | ||
| x3 |
| −10 | ||
| 3 | ||
| 0 |
1.4.22 |
| 0 |
| 0 |
| −4 |
| 0 |
| −3 |
| −8 |
| 5 |
| −2 |
| −12 |
1.4.22 |
| 0 |
| 0 |
| −2 |
| 0 |
| −6 |
| −6 |
| 6 |
| −4 |
| −4 |
1.4.22 |
| 0 |
| 0 |
| −4 |
| 0 |
| −4 |
| −8 |
| 4 |
| −3 |
| 12 |
1.4.22 |
| 0 |
| 0 |
| −2 |
| 0 |
| −3 |
| 6 |
| 6 |
| −2 |
| −4 |
1.4.22 |
| 0 |
| 0 |
| −5 |
| 0 |
| −2 |
| −10 |
| 3 |
| −5 |
| −15 |
1.4.26 |
1.4.26 |
1.4.27 |
| A | b |
| A | b |
| A | b |
| A | b |
| A | b |
1.4.33 |
1.4.33 |
1.4.41 |
1.4.41 |
1.4.44 |
1.4.44 |
1.5.1 |
2x1−9x2+11x3 | = | 0 |
−2x1−11x2+4x3 | = | 0 |
4x1+2x2+7x3 | = | 0 |
1.5.1 |
2x1−9x2+11x3 | = | 0 |
−2x1−11x2+4x3 | = | 0 |
4x1+2x2+7x3 | = | 0 |
1.5.1 |
6x1−3x2+15x3 | = | 0 |
−6x1−9x2−6x3 | = | 0 |
12x1+6x2+21x3 | = | 0 |
1.5.1 |
2x1−5x2+8x3 | = | 0 |
−2x1−7x2+x3 | = | 0 |
4x1+2x2+7x3 | = | 0 |
1.5.1 |
4x1−2x2+10x3 | = | 0 |
−4x1−2x2−7x3 | = | 0 |
8x1+4x2+14x3 | = | 0 |
1.5.2 |
x1 | − | 3x2 | + | 6x3 | = | 0 |
−3x1 | + | 4x2 | − | 10x3 | = | 0 |
x1 | + | 2x2 | + | 10x3 | = | 0 |
1.5.2 |
x1 | − | 2x2 | + | 6x3 | = | 0 |
−4x1 | + | 3x2 | − | 15x3 | = | 0 |
x1 | + | 3x2 | + | 10x3 | = | 0 |
1.5.5 |
3x1+3x2+6x3 | = | 0 | where the solution set is x=
| |||
−6x1−6x2−12x3 | = | 0 | ||||
−7x2+14x3 | = | 0 |
| −4 | ||
| 2 | ||
| 1 |
1.5.5 |
2x1+2x2+4x3 | = | 0 | where the solution set is x=
| |||
−4x1−4x2−8x3 | = | 0 | ||||
−6x2+18x3 | = | 0 |
| −5 | ||
| 3 | ||
| 1 |
1.5.7 |
| 1 | 4 | −5 | 8 | ||
| 0 | 1 | −5 | 5 |
| −15 | ||
| 5 | ||
| 1 | ||
| 0 |
| 12 | ||
| −5 | ||
| 0 | ||
| 1 |
1.5.7 |
| 1 | 3 | −5 | 5 | ||
| 0 | 1 | −2 | 8 |
| −1 | ||
| 2 | ||
| 1 | ||
| 0 |
| 19 | ||
| −8 | ||
| 0 | ||
| 1 |
1.5.8 |
| 1 | −4 | −9 | 4 | ||
| 0 | 1 | 4 | −2 |
| −7 | ||
| −4 | ||
| 1 | ||
| 0 |
| 4 | ||
| 2 | ||
| 0 | ||
| 1 |
1.5.8 |
| 1 | −2 | −5 | 3 | ||
| 0 | 1 | 3 | −4 |
| −1 | ||
| −3 | ||
| 1 | ||
| 0 |
| 5 | ||
| 4 | ||
| 0 | ||
| 1 |
1.5.8 |
| 1 | −2 | −7 | 2 | ||
| 0 | 1 | 5 | −2 |
| −3 | ||
| −5 | ||
| 1 | ||
| 0 |
| 2 | ||
| 2 | ||
| 0 | ||
| 1 |
1.5.8 |
| 1 | −4 | −7 | 5 | ||
| 0 | 1 | 2 | −7 |
| −1 | ||
| −2 | ||
| 1 | ||
| 0 |
| 23 | ||
| 7 | ||
| 0 | ||
| 1 |
1.5.9 |
| 4 | −16 | 8 | ||
| −1 | 4 | −2 |
| 4 | ||
| 1 | ||
| 0 |
| −2 | ||
| 0 | ||
| 1 |
1.5.19 |
3x1+3x2+6x3 | = | 12 | 3x1+3x2+6x3 | = | 0 | |
−9x1−9x2−18x3 | = | −36 | −9x1−9x2−18x3 | = | 0 | |
−4x2−4x3 | = | 16 | −4x2−4x3 | = | 0 |
| x1 |
| x2 |
| x3 |
| 8 | ||
| −4 | ||
| 0 |
| −1 | ||
| −1 | ||
| 1 |
1.5.19 |
2x1+2x2+4x3 | = | 8 | 2x1+2x2+4x3 | = | 0 | |
−6x1−6x2−12x3 | = | −24 | −6x1−6x2−12x3 | = | 0 | |
−7x2+14x3 | = | 14 | −7x2+14x3 | = | 0 |
| x1 |
| x2 |
| x3 |
| 6 | ||
| −2 | ||
| 0 |
| −4 | ||
| 2 | ||
| 1 |
1.5.19 |
2x1+2x2+4x3 | = | 8 | 2x1+2x2+4x3 | = | 0 | |
−4x1−4x2−8x3 | = | −16 | −4x1−4x2−8x3 | = | 0 | |
−3x2+6x3 | = | 15 | −3x2+6x3 | = | 0 |
| x1 |
| x2 |
| x3 |
| 9 | ||
| −5 | ||
| 0 |
| −4 | ||
| 2 | ||
| 1 |
1.5.19 |
2x1+2x2+4x3 | = | 8 | 2x1+2x2+4x3 | = | 0 | |
−4x1−4x2−8x3 | = | −16 | −4x1−4x2−8x3 | = | 0 | |
−3x2+6x3 | = | 15 | −3x2+6x3 | = | 0 |
| x1 |
| x2 |
| x3 |
| 9 | ||
| −5 | ||
| 0 |
| −4 | ||
| 2 | ||
| 1 |
1.5.28 |
1.5.28 |
1.5.28 |
1.5.31 |
1.5.31 |
1.5.35 |
1.5.35 |
1.5.40 |
1.5.40 |
1.5.44 |
1.5.44 |
1.6.1 |
Suppose an economy has only two sectors: Goods and Services. Each year, Goods sells 75% of its outputs to Services and keeps the rest, while Services sells 66% of its output to Goods and retains the rest. Find equilibrium prices for the annual outputs of the Goods and Services sectors that make each sector's income match its expenditures. | 0.34 0.75 0.66 0.25 |
1.6.1 |
Suppose an economy has only two sectors: Goods and Services. Each year, Goods sells 80% of its outputs to Services and keeps the rest, while Services sells 65% of its output to Goods and retains the rest. Find equilibrium prices for the annual outputs of the Goods and Services sectors that make each sector's income match its expenditures. | 0.35 0.8 0.65 0.2 |
1.6.1 |
Suppose an economy has only two sectors: Goods and Services. Each year, Goods sells 75% of its outputs to Services and keeps the rest, while Services sells 60% of its output to Goods and retains the rest. Find equilibrium prices for the annual outputs of the Goods and Services sectors that make each sector's income match its expenditures. | 0.4 0.75 0.6 0.25 |
1.6.2 |
1.6.2 |
1.6.3 |
Distribution of Output from: | |||
|---|---|---|---|
Chemicals | Fuels | Machinery | Purchased by: |
0.10 | 0.80 | 0.30 | Chemicals |
0.30 | 0.10 | 0.20 | Fuels |
0.60 | 0.10 | 0.50 | Machinery |
| 0.90 | −0.80 | −0.30 | 0 | ||
| −0.30 | 0.90 | −0.20 | 0 | ||
| −0.60 | −0.10 | 0.50 | 0 |
1.6.3 |
Distribution of Output from: | |||
|---|---|---|---|
Chemicals | Fuels | Machinery | Purchased by: |
0.20 | 0.70 | 0.50 | Chemicals |
0.20 | 0.10 | 0.30 | Fuels |
0.60 | 0.20 | 0.20 | Machinery |
| 0.8 | −0.7 | −0.5 | 0 | ||
| −0.2 | 0.9 | −0.3 | 0 | ||
| −0.6 | −0.2 | 0.8 | 0 |
1.6.4 |
Distribution of Output from: | ||||
|---|---|---|---|---|
Mining | Lumber | Energy | Transportation | Purchased by: |
0.15 | 0.15 | 0.30 | 0.10 | Mining |
0.15 | 0.15 | 0.10 | 0.20 | Lumber |
0.70 | 0.50 | 0.35 | 0.50 | Energy |
0.00 | 0.20 | 0.25 | 0.20 | Transportation |
1.6.4 |
Distribution of Output from: | ||||
|---|---|---|---|---|
Mining | Lumber | Energy | Transportation | Purchased by: |
0.40 | 0.10 | 0.30 | 0.25 | Mining |
0.10 | 0.25 | 0.10 | 0.10 | Lumber |
0.50 | 0.40 | 0.40 | 0.50 | Energy |
0.00 | 0.25 | 0.20 | 0.15 | Transportation |
1.6.5 |
1.6.5 |
1.6.5 |
1.6.5 |
1.6.7 |
1.6.9 |
1.6.9 |
1.6.9 |
1.6.9 |
1.7.5 |
Determine if the columns of the matrix form a linearly independent set. Justify your answer. |
|
| 0 | −8 | 16 | 0 | ||
| 3 | 1 | −14 | 0 | ||
| −1 | 5 | −4 | 0 | ||
| 1 | −5 | −2 | 0 |
1.7.5 |
Determine if the columns of the matrix form a linearly independent set. Justify your answer. |
|
| 0 | −3 | 9 | 0 | ||
| 2 | 1 | −7 | 0 | ||
| −1 | 4 | −8 | 0 | ||
| 1 | −4 | −2 | 0 |
1.7.5 |
Determine if the columns of the matrix form a linearly independent set. Justify your answer. |
|
| 0 | −8 | 16 | 0 | ||
| 3 | 1 | −14 | 0 | ||
| −1 | 5 | −6 | 0 | ||
| 1 | −5 | −2 | 0 |
1.7.5 |
Determine if the columns of the matrix form a linearly independent set. Justify your answer. |
|
| 0 | −3 | 9 | 0 | ||
| 2 | 1 | −7 | 0 | ||
| −1 | 4 | −6 | 0 | ||
| 1 | −4 | −2 | 0 |
1.7.5 |
Determine if the columns of the matrix form a linearly independent set. Justify your answer. |
|
| 0 | −3 | 9 | 0 | ||
| 2 | 1 | −7 | 0 | ||
| −1 | 4 | −7 | 0 | ||
| 1 | −4 | −2 | 0 |
1.7.6 |
Determine if the columns of the matrix form a linearly independent set. Justify your answer. |
|
| −4 | −3 | 0 | 0 | ||
| 0 | −1 | 6 | 0 | ||
| 1 | 1 | −6 | 0 | ||
| 2 | 1 | −12 | 0 |
1.7.6 |
Determine if the columns of the matrix form a linearly independent set. Justify your answer. |
|
| −4 | −3 | 0 | 0 | ||
| 0 | −1 | 5 | 0 | ||
| 1 | 1 | −5 | 0 | ||
| 2 | 1 | −10 | 0 |
1.7.8 |
| 1 | −2 | 3 |
| −2 | 4 | 3 |
1.7.8 |
| 1 | −2 | 3 | 4 |
| −2 | 4 | −6 | 4 |
| 0 | 1 | −1 | 5 |
1.7.13 |
| 1 |
| 3 |
| −2 |
| −4 |
| −11 |
| 8 |
| 3 |
| h |
| −6 |
1.7.13 |
| 1 |
| 5 |
| −2 |
| −2 |
| −9 |
| 4 |
| 2 |
| h |
| −4 |
1.7.14 |
| 1 |
| −3 |
| −6 |
| −4 |
| 13 |
| 8 |
| 4 |
| 1 |
| h |
1.7.14 |
| 1 |
| −2 |
| −4 |
| −3 |
| 7 |
| 6 |
| 3 |
| 1 |
| h |
1.7.16 |
| 6 |
| −12 |
| 9 |
| 4 |
| −8 |
| 6 |
1.7.16 |
| 6 |
| −4 |
| 2 |
| 9 |
| −6 |
| 3 |
1.7.17 |
| 4 |
| 2 |
| −3 |
| 0 |
| 0 |
| 0 |
| −2 |
| 4 |
| 4 |
1.7.17 |
| 6 |
| 2 |
| −3 |
| 0 |
| 0 |
| 0 |
| −4 |
| 3 |
| 2 |
1.7.19 |
| −8 |
| 16 |
| 4 |
| 2 |
| −4 |
| 1 |
1.7.19 |
| 10 |
| −20 |
| −5 |
| −2 |
| 4 |
| −1 |
1.7.19 |
| 8 |
| −12 |
| −4 |
| −2 |
| 3 |
| −1 |
1.7.23 |
1.7.23 |
1.7.24 |
1.7.24 |
1.7.24 |
1.7.24 |
1.7.24 |
1.7.25 |
1.7.25 |
1.7.32 |
| a1 | a2 | a3 |
| X | 0 | 0 |
| 0 | 0 | 0 |
| 0 | 0 | 0 |
| 0 | X | * |
| 0 | 0 | X |
| 0 | 0 | 0 |
| 0 | 0 | X |
| 0 | 0 | 0 |
| 0 | 0 | 0 |
| X | * | * |
| 0 | X | * |
| 0 | 0 | X |
1.7.32 |
| a1 | a2 | a3 |
| 0 | X | * |
| 0 | 0 | X |
| 0 | 0 | 0 |
| 0 | 0 | 0 |
| X | * | * |
| X | * | * |
| X | * | * |
| 0 | X | X |
| X | * | * |
| 0 | X | * |
| 0 | 0 | X |
| 0 | 0 | 0 |
| X | 0 | 0 |
| 0 | 0 | 0 |
| 0 | 0 | 0 |
| 0 | 0 | 0 |
1.7.42 |
| 1 |
| 1 |
| 1 |
| 2 |
| 2 |
| 2 |
| 1 |
| 0 |
| 0 |
1.7.42 |
| 1 |
| 1 |
| 1 |
| 2 |
| 2 |
| 2 |
| 1 |
| 0 |
| 0 |
1.8.1 |
| 2 | 0 | ||
| 0 | 2 |
| 2 | ||
| −6 |
| a | ||
| b |
| 4 | ||
| −12 |
| 2a | ||
| 2b |
1.8.1 |
| 3 | 0 | ||
| 0 | 3 |
| 4 | ||
| −5 |
| a | ||
| b |
| 12 | ||
| −15 |
| 3a | ||
| 3b |
1.8.4 |
| 1 | −3 | 4 | ||
| 0 | 1 | −5 | ||
| 4 | −13 | 16 |
| −4 | ||
| −8 | ||
| −3 |
| −39 | ||
| −13 | ||
| −1 |
1.8.4 |
| 1 | −4 | 4 | ||
| 0 | 1 | −4 | ||
| 2 | −9 | 8 |
| −4 | ||
| −1 | ||
| −3 |
| −20 | ||
| −5 | ||
| −1 |
1.8.4 |
| 1 | −2 | 3 | ||
| 0 | 1 | −3 | ||
| 3 | −7 | 9 |
| −4 | ||
| −6 | ||
| −3 |
| −19 | ||
| −9 | ||
| −1 |
1.8.5 |
| 1 | −7 | −19 | ||
| −4 | 21 | 55 |
| −3 | ||
| 5 |
| 4 | ||
| 1 | ||
| 0 |
1.8.5 |
| 1 | −3 | −10 | ||
| −4 | 6 | 16 |
| −1 | ||
| −2 |
| 2 | ||
| 1 | ||
| 0 |
1.8.6 |
| 1 | 3 | 6 | ||
| 4 | 15 | 27 | ||
| 0 | 1 | 1 | ||
| −4 | −13 | −25 |
| 16 | ||
| 73 | ||
| 3 | ||
| −67 |
| 7 | ||
| 3 | ||
| 0 |
1.8.6 |
| 1 | 3 | 7 | ||
| 2 | 9 | 17 | ||
| 0 | 1 | 1 | ||
| −2 | −7 | −15 |
| 12 | ||
| 30 | ||
| 2 | ||
| −26 |
| 6 | ||
| 2 | ||
| 0 |
1.8.7 |
1.8.7 |
1.8.10 |
| 1 | 2 | 10 | 1 | ||
| 1 | 0 | 2 | −4 | ||
| 0 | 1 | 4 | 5 | ||
| −1 | 7 | 26 | 24 |
| −2 | ||
| −4 | ||
| 1 | ||
| 0 |
1.8.10 |
| 1 | 2 | 6 | −1 | ||
| 1 | 0 | 2 | −4 | ||
| 0 | 1 | 2 | 3 | ||
| −2 | 4 | 4 | 10 |
| −2 | ||
| −2 | ||
| 1 | ||
| 0 |
1.8.10 |
| 1 | 2 | 6 | −1 | ||
| 1 | 0 | 2 | −4 | ||
| 0 | 1 | 2 | 3 | ||
| −3 | 1 | −4 | 8 |
| −2 | ||
| −2 | ||
| 1 | ||
| 0 |
1.8.10 |
| 1 | 3 | 13 | 3 | ||
| 1 | 0 | 4 | −5 | ||
| 0 | 1 | 3 | 4 | ||
| −1 | 6 | 14 | 13 |
| −4 | ||
| −3 | ||
| 1 | ||
| 0 |
1.8.10 |
| 1 | 4 | 20 | 11 | ||
| 1 | 0 | 4 | −4 | ||
| 0 | 1 | 4 | 5 | ||
| −2 | 1 | −4 | 5 |
| −4 | ||
| −4 | ||
| 1 | ||
| 0 |
1.8.12 |
| −13 | ||
| 1 | ||
| −3 | ||
| 19 |
| 1 | 4 | 12 | 5 | ||
| 1 | 0 | 4 | −5 | ||
| 0 | 1 | 2 | 3 | ||
| −3 | −4 | −20 | −11 |
1.8.12 |
| −3 | ||
| 2 | ||
| −2 | ||
| 3 |
| 1 | 2 | 7 | 0 | ||
| 1 | 0 | 3 | −5 | ||
| 0 | 1 | 2 | 3 | ||
| −4 | −1 | −14 | 1 |
1.8.12 |
| −13 | ||
| 1 | ||
| −3 | ||
| 19 |
| 1 | 4 | 20 | 15 | ||
| 1 | 0 | 4 | −3 | ||
| 0 | 1 | 4 | 5 | ||
| −3 | −4 | −28 | −21 |
1.8.17 |
| 3 |
| 5 |
| 4 |
| 1 |
| 3 |
| 3 |
| −1 |
| 3 |
| 12 | ||
| 3 |
| −4 | ||
| 12 |
| 8 | ||
| 15 |
1.8.17 |
| 3 |
| 4 |
| 6 |
| 1 |
| 3 |
| 3 |
| −1 |
| 3 |
| 12 | ||
| 2 |
| −4 | ||
| 12 |
| 8 | ||
| 14 |
1.8.20 |
| x1 | ||
| x2 |
| 8 | ||
| −2 |
| 9 | ||
| −6 |
| 8 | 9 | ||
| −2 | −6 |
1.8.20 |
| x1 | ||
| x2 |
| −6 | ||
| 5 |
| 4 | ||
| −7 |
| −6 | 4 | ||
| 5 | −7 |
1.8.21 |
1.8.21 |
1.8.22 |
1.8.22 |
1.8.22 |
1.8.22 |
1.8.23 |
1.8.23 |
1.8.25 |
1.8.25 |
1.8.28 |
1.8.28 |
1.8.29 |
1.8.29 |
1.8.29 |
1.8.31 |
| 1 | 0 | ||
| 0 | −1 |
1.8.31 |
| 1 | 0 | ||
| 0 | −1 |
1.8.39 |
1.8.39 |
1.9.1 |
| 2 | −9 | ||
| 1 | 5 | ||
| 2 | 0 | ||
| 1 | 0 |
1.9.1 |
| 2 | −4 | ||
| 1 | 7 | ||
| 2 | 0 | ||
| 1 | 0 |
1.9.1 |
| 4 | −5 | ||
| 1 | 3 | ||
| 4 | 0 | ||
| 1 | 0 |
1.9.1 |
| 8 | −7 | ||
| 1 | 5 | ||
| 8 | 0 | ||
| 1 | 0 |
1.9.15 |
| ? | ? | ? |
| ? | ? | ? |
| ? | ? | ? |
| x1 |
| x2 |
| x3 |
| 3x1−5x2 |
| 7x1−4x3 |
| −4x2+5x3 |
| 3 | −5 | 0 |
| 7 | 0 | −4 |
| 0 | −4 | 5 |
| x1 |
| x2 |
| x3 |
| 3x1−5x2 |
| 7x1−4x3 |
| −4x2+5x3 |
1.9.15 |
| ? | ? | ? |
| ? | ? | ? |
| ? | ? | ? |
| x1 |
| x2 |
| x3 |
| 4x1−3x2 |
| x1−4x3 |
| −7x2+8x3 |
| 4 | −3 | 0 |
| 1 | 0 | −4 |
| 0 | −7 | 8 |
| x1 |
| x2 |
| x3 |
| 4x1−3x2 |
| x1−4x3 |
| −7x2+8x3 |
1.9.17 |
| 1 | 4 | 0 | 0 | ||
| 0 | 0 | 0 | 0 | ||
| 0 | 8 | 0 | 1 | ||
| 0 | 1 | 0 | −1 |
1.9.17 |
| 1 | 8 | 0 | 0 | ||
| 0 | 0 | 0 | 0 | ||
| 0 | 5 | 0 | 1 | ||
| 0 | 1 | 0 | −1 |
1.9.17 |
| 1 | 9 | 0 | 0 | ||
| 0 | 0 | 0 | 0 | ||
| 0 | 6 | 0 | 1 | ||
| 0 | 1 | 0 | −1 |
1.9.17 |
| 1 | 5 | 0 | 0 | ||
| 0 | 0 | 0 | 0 | ||
| 0 | 9 | 0 | 1 | ||
| 0 | 1 | 0 | −1 |
1.9.19 |
| 1 | −2 | 7 | ||
| 0 | 1 | −8 |
1.9.20 |
| −4 | 1 | −1 | 1 |
1.9.20 |
| 3 | 1 | −2 | −4 |
1.9.24 |
1.9.24 |
1.9.24 |
1.9.24 |
1.9.29 |
1.9.29 |
1.9.29 |
1.9.29 |
1.9.31 |
1.9.31 |
1.9.31 |
1.9.31 |
1.9.33 |
1.9.33 |
1.9.35 |
1.9.35 |
1.9.45 |
Let T be the linear transformation whose standard matrix is given. Decide if T is a one-to-one mapping. Justify your answer. |
|
1.9.45 |
Let T be the linear transformation whose standard matrix is given. Decide if T is a one-to-one mapping. Justify your answer. |
|
1.9.45 |
Let T be the linear transformation whose standard matrix is given. Decide if T is a one-to-one mapping. Justify your answer. |
|
1.10.7 |
| 15 | −9 | 0 | −3 | ||
| −9 | 17 | −6 | 0 | ||
| 0 | −6 | 15 | −6 | ||
| −3 | 0 | −6 | 13 |
| I1 | ||
| I2 | ||
| I3 | ||
| I4 |
| 41 | ||
| 25 | ||
| 24 | ||
| −11 |
1.10.7 |
| 12 | −8 | 0 | −3 | ||
| −8 | 16 | −6 | 0 | ||
| 0 | −6 | 14 | −4 | ||
| −3 | 0 | −4 | 12 |
| I1 | ||
| I2 | ||
| I3 | ||
| I4 |
| 39 | ||
| 28 | ||
| 18 | ||
| −12 |
1.10.8 |
Write a matrix equation that determines the loop currents shown to the right. | I1 I2 I4 I3 I5 |
| 12 | −2 | 0 | −2 | −4 | ||
| −2 | 13 | −4 | 0 | −4 | ||
| 0 | −4 | 17 | −5 | −5 | ||
| −2 | 0 | −5 | 11 | −3 | ||
| −4 | −4 | −5 | −3 | 16 |
| I1 | ||
| I2 | ||
| I3 | ||
| I4 | ||
| I5 |
| 51 | ||
| −38 | ||
| 24 | ||
| −48 | ||
| 0 |
1.10.8 |
Write a matrix equation that determines the loop currents shown to the right. | I1 I2 I4 I3 I5 |
| 12 | −2 | 0 | −2 | −4 | ||
| −2 | 11 | −4 | 0 | −4 | ||
| 0 | −4 | 13 | −3 | −3 | ||
| −2 | 0 | −3 | 12 | −4 | ||
| −4 | −4 | −3 | −4 | 15 |
| I1 | ||
| I2 | ||
| I3 | ||
| I4 | ||
| I5 |
| 51 | ||
| −35 | ||
| 29 | ||
| −40 | ||
| 0 |
1.10.8 |
Write a matrix equation that determines the loop currents shown to the right. | I1 I2 I4 I3 I5 |
| 8 | −1 | 0 | −2 | −3 | ||
| −1 | 9 | −4 | 0 | −2 | ||
| 0 | −4 | 13 | −3 | −4 | ||
| −2 | 0 | −3 | 9 | −2 | ||
| −3 | −2 | −4 | −2 | 11 |
| I1 | ||
| I2 | ||
| I3 | ||
| I4 | ||
| I5 |
| 52 | ||
| −38 | ||
| 24 | ||
| −45 | ||
| 0 |
1.10.8 |
Write a matrix equation that determines the loop currents shown to the right. | I1 I2 I4 I3 I5 |
| 9 | −1 | 0 | −1 | −3 | ||
| −1 | 9 | −3 | 0 | −3 | ||
| 0 | −3 | 14 | −4 | −5 | ||
| −1 | 0 | −4 | 9 | −3 | ||
| −3 | −3 | −5 | −3 | 14 |
| I1 | ||
| I2 | ||
| I3 | ||
| I4 | ||
| I5 |
| 55 | ||
| −33 | ||
| 28 | ||
| −40 | ||
| 0 |
1.10.12 |
Cars Rented From: | ||||||||||||||
Airport | East | West | Returned To: | |||||||||||
| Airport | |||||||||||||
East | ||||||||||||||
West | ||||||||||||||
1.10.12 |
Cars Rented From: | ||||||||||||||
Airport | East | West | Returned To: | |||||||||||
| Airport | |||||||||||||
East | ||||||||||||||
West | ||||||||||||||
2.1.1 |
| 2 | 0 | −1 | ||
| 4 | −6 | 4 |
| 7 | −4 | 2 | ||
| 1 | −5 | −2 |
| 2 | 3 | ||
| −2 | 2 |
| 4 | 6 | ||
| −2 | 5 |
| −6 | 0 | 3 | ||
| −12 | 18 | −12 |
| 1 | −4 | 5 | ||
| −11 | 13 | −14 |
| 2 | 27 | ||
| −12 | −2 |
2.1.1 |
| 3 | 0 | −1 | ||
| 3 | −6 | 3 |
| 6 | −6 | 1 | ||
| 2 | −5 | −3 |
| 1 | 3 | ||
| −1 | 1 |
| 4 | 6 | ||
| −2 | 4 |
| −6 | 0 | 2 | ||
| −6 | 12 | −6 |
| 0 | −6 | 3 | ||
| −4 | 7 | −9 |
| −2 | 18 | ||
| −6 | −2 |
2.1.1 |
| 1 | 0 | −1 | ||
| 4 | −6 | 3 |
| 8 | −4 | 2 | ||
| 2 | −4 | −2 |
| 2 | 3 | ||
| −1 | 2 |
| 4 | 5 | ||
| −1 | 4 |
| −3 | 0 | 3 | ||
| −12 | 18 | −9 |
| 5 | −4 | 5 | ||
| −10 | 14 | −11 |
| 5 | 22 | ||
| −6 | 3 |
2.1.2 |
| 1 | 0 | −3 | ||
| 3 | −6 | 4 |
| 7 | −4 | 2 | ||
| 2 | −3 | −3 |
| 1 | 3 | ||
| −2 | 1 |
| 3 | 4 | ||
| −2 | 5 |
| −5 | ||
| 2 |
| 22 | −12 | 3 | ||
| 9 | −15 | −5 |
| 29 | −24 | −6 | ||
| −4 | −7 | −19 |
2.1.2 |
| 1 | 0 | −1 | ||
| 5 | −4 | 2 |
| 8 | −4 | 1 | ||
| 2 | −4 | −3 |
| 1 | 3 | ||
| −2 | 1 |
| 4 | 6 | ||
| −2 | 4 |
| −5 | ||
| 4 |
| 33 | −16 | 3 | ||
| 13 | −20 | −10 |
| 44 | −40 | −14 | ||
| −8 | −8 | −14 |
2.1.2 |
| 2 | 0 | −1 | ||
| 4 | −5 | 3 |
| 6 | −4 | 2 | ||
| 2 | −3 | −4 |
| 2 | 3 | ||
| −1 | 2 |
| 2 | 6 | ||
| −2 | 4 |
| −5 | ||
| 4 |
| 20 | −12 | 5 | ||
| 10 | −14 | −9 |
| 24 | −26 | −20 | ||
| −4 | −4 | −20 |
2.1.4 |
| 6 | −2 | 3 | ||
| −3 | 4 | −6 | ||
| −3 | 1 | 3 |
| 3 | −2 | 3 | ||
| −3 | 1 | −6 | ||
| −3 | 1 | 0 |
| 18 | −6 | 9 | ||
| −9 | 12 | −18 | ||
| −9 | 3 | 9 |
2.1.4 |
| 6 | −1 | 2 | ||
| −4 | 3 | −5 | ||
| −2 | 1 | 1 |
| 3 | −1 | 2 | ||
| −4 | 0 | −5 | ||
| −2 | 1 | −2 |
| 18 | −3 | 6 | ||
| −12 | 9 | −15 | ||
| −6 | 3 | 3 |
2.1.4 |
| 6 | −2 | 2 | ||
| −5 | 2 | −7 | ||
| −4 | 2 | 2 |
| 1 | −2 | 2 | ||
| −5 | −3 | −7 | ||
| −4 | 2 | −3 |
| 30 | −10 | 10 | ||
| −25 | 10 | −35 | ||
| −20 | 10 | 10 |
2.1.5 |
| −2 | 3 | ||
| 1 | 4 | ||
| 5 | −4 |
| 5 | −3 | ||
| −1 | 3 |
| −2 | 3 | ||
| 1 | 4 | ||
| 5 | −4 |
| 5 | ||
| −1 |
| −13 | ||
| 1 | ||
| 29 |
| −2 | 3 | ||
| 1 | 4 | ||
| 5 | −4 |
| −3 | ||
| 3 |
| 15 | ||
| 9 | ||
| −27 |
| −13 | 15 | ||
| 1 | 9 | ||
| 29 | −27 |
2.1.5 |
| −1 | 3 | ||
| 3 | 3 | ||
| 4 | −3 |
| 5 | −1 | ||
| −1 | 2 |
| −1 | 3 | ||
| 3 | 3 | ||
| 4 | −3 |
| 5 | ||
| −1 |
| −8 | ||
| 12 | ||
| 23 |
| −1 | 3 | ||
| 3 | 3 | ||
| 4 | −3 |
| −1 | ||
| 2 |
| 7 | ||
| 3 | ||
| −10 |
| −8 | 7 | ||
| 12 | 3 | ||
| 23 | −10 |
2.1.5 |
| −2 | 2 | ||
| 1 | 5 | ||
| 6 | −3 |
| 4 | −3 | ||
| −3 | 3 |
| −2 | 2 | ||
| 1 | 5 | ||
| 6 | −3 |
| 4 | ||
| −3 |
| −14 | ||
| −11 | ||
| 33 |
| −2 | 2 | ||
| 1 | 5 | ||
| 6 | −3 |
| −3 | ||
| 3 |
| 12 | ||
| 12 | ||
| −27 |
| −14 | 12 | ||
| −11 | 12 | ||
| 33 | −27 |
2.1.7 |
2.1.7 |
2.1.8 |
2.1.8 |
2.1.9 |
| 2 | 3 |
| −1 | 1 |
| 2 | 9 |
| −3 | k |
2.1.9 |
| 3 | 4 |
| −1 | 1 |
| 2 | 8 |
| −2 | k |
2.1.9 |
| 3 | 2 |
| −1 | 2 |
| 2 | 6 |
| −3 | k |
2.1.9 |
| 4 | 2 |
| −1 | 2 |
| 2 | 4 |
| −2 | k |
2.1.10 |
| −4 | 6 | ||
| 8 | −12 |
| 6 | 2 | ||
| 4 | 7 |
| 3 | −1 | ||
| 2 | 5 |
| −4(6)+(−12)(7) | −4(2)+(−12)(4) | ||
| 8(6)+6(7) | 8(2)+6(4) |
| 8(6)+6(4) | 8(2)+6(7) | ||
| −4(6)+(−12)(4) | −4(2)+(−12)(7) |
| −4(6)+6(4) | −4(2)+6(7) | ||
| 8(6)+(−12)(4) | 8(2)+(−12)(7) |
| −4(2)+6(4) | −4(6)+6(7) | ||
| 8(2)+(−12)(4) | 8(6)+(−12)(7) |
| 8(3)+6(2) | 8(−1)+6(5) | ||
| −4(3)+(−12)(2) | −4(−1)+(−12)(5) |
| −4(3)+6(2) | −4(−1)+6(5) | ||
| 8(3)+(−12)(2) | 8(−1)+(−12)(5) |
| −4(−1)+6(2) | −4(3)+6(5) | ||
| 8(−1)+(−12)(2) | 8(3)+(−12)(5) |
| −4(3)+(−12)(5) | −4(−1)+(−12)(2) | ||
| 8(3)+6(5) | 8(−1)+6(2) |
| 0 | 34 | ||
| 0 | −68 |
2.1.10 |
| 4 | −6 | ||
| −8 | 12 |
| 8 | 4 | ||
| 6 | 9 |
| 2 | −2 | ||
| 2 | 5 |
| 4(4)+(−6)(6) | 4(8)+(−6)(9) | ||
| −8(4)+12(6) | −8(8)+12(9) |
| −8(8)+(−6)(6) | −8(4)+(−6)(9) | ||
| 4(8)+12(6) | 4(4)+12(9) |
| 4(8)+12(9) | 4(4)+12(6) | ||
| −8(8)+(−6)(9) | −8(4)+(−6)(6) |
| 4(8)+(−6)(6) | 4(4)+(−6)(9) | ||
| −8(8)+12(6) | −8(4)+12(9) |
| 4(−2)+(−6)(2) | 4(2)+(−6)(5) | ||
| −8(−2)+12(2) | −8(2)+12(5) |
| 4(2)+12(5) | 4(−2)+12(2) | ||
| −8(2)+(−6)(5) | −8(−2)+(−6)(2) |
| −8(2)+(−6)(2) | −8(−2)+(−6)(5) | ||
| 4(2)+12(2) | 4(−2)+12(5) |
| 4(2)+(−6)(2) | 4(−2)+(−6)(5) | ||
| −8(2)+12(2) | −8(−2)+12(5) |
| −4 | −38 | ||
| 8 | 76 |
2.1.10 |
| −4 | 6 | ||
| 8 | −12 |
| 5 | 4 | ||
| 7 | 9 |
| −1 | −2 | ||
| 3 | 5 |
| −4(5)+6(7) | −4(4)+6(9) | ||
| 8(5)+(−12)(7) | 8(4)+(−12)(9) |
| −4(5)+(−12)(9) | −4(4)+(−12)(7) | ||
| 8(5)+6(9) | 8(4)+6(7) |
| 8(5)+6(7) | 8(4)+6(9) | ||
| −4(5)+(−12)(7) | −4(4)+(−12)(9) |
| −4(4)+6(7) | −4(5)+6(9) | ||
| 8(4)+(−12)(7) | 8(5)+(−12)(9) |
| −4(−1)+6(3) | −4(−2)+6(5) | ||
| 8(−1)+(−12)(3) | 8(−2)+(−12)(5) |
| −4(−2)+6(3) | −4(−1)+6(5) | ||
| 8(−2)+(−12)(3) | 8(−1)+(−12)(5) |
| 8(−1)+6(3) | 8(−2)+6(5) | ||
| −4(−1)+(−12)(3) | −4(−2)+(−12)(5) |
| −4(−1)+(−12)(5) | −4(−2)+(−12)(3) | ||
| 8(−1)+6(5) | 8(−2)+6(3) |
| 22 | 38 | ||
| −44 | −76 |
2.1.11 |
| 1 | 1 | 1 |
| 1 | 4 | 5 |
| 1 | 5 | 6 |
| 6 | 0 | 0 |
| 0 | 5 | 0 |
| 0 | 0 | 2 |
| 6 | 5 | 2 | ||
| 6 | 20 | 10 | ||
| 6 | 25 | 12 |
| 6 | 6 | 6 | ||
| 5 | 20 | 25 | ||
| 2 | 10 | 12 |
2.1.11 |
| 1 | 1 | 1 |
| 1 | 5 | 7 |
| 1 | 7 | 6 |
| 5 | 0 | 0 |
| 0 | 3 | 0 |
| 0 | 0 | 2 |
| 5 | 3 | 2 | ||
| 5 | 15 | 14 | ||
| 5 | 21 | 12 |
| 5 | 5 | 5 | ||
| 3 | 15 | 21 | ||
| 2 | 14 | 12 |
2.1.11 |
| 1 | 1 | 1 |
| 1 | 4 | 6 |
| 1 | 6 | 7 |
| 7 | 0 | 0 |
| 0 | 4 | 0 |
| 0 | 0 | 3 |
| 7 | 4 | 3 | ||
| 7 | 16 | 18 | ||
| 7 | 24 | 21 |
| 7 | 7 | 7 | ||
| 4 | 16 | 24 | ||
| 3 | 18 | 21 |
2.1.12 |
| 3 | −6 | ||
| −5 | 10 |
| 2 | 4 | ||
| 1 | 2 |
2.1.12 |
| 5 | −15 | ||
| −3 | 9 |
| 6 | 3 | ||
| 2 | 1 |
2.1.16 |
| b1 | b2 | b3 |
| Ab1+Ab2+Ab3 |
| Ab1+Ab2+Ab3 |
| Ab1+Ab2+Ab3 |
| b1 | b2 | ... | bp |
| Ab1+Ab2+...+Abp |
2.1.16 |
| b1 | b2 | b3 |
| Ab1+Ab2+Ab3 |
| Ab1+Ab2+Ab3 |
| b1 | b2 | ... | bp |
| Ab1+Ab2+...+Abp |
| Ab1+Ab2+Ab3 |
2.1.19 |
2.1.19 |
2.1.20 |
2.1.20 |
2.1.20 |
2.1.21 |
2.1.21 |
2.1.21 |
2.1.22 |
2.1.22 |
2.1.22 |
2.1.26 |
2.1.26 |
2.1.47 |
| 0 | 9 | 0 | 0 | 0 | ||
| 0 | 0 | 9 | 0 | 0 | ||
| 0 | 0 | 0 | 9 | 0 | ||
| 0 | 0 | 0 | 0 | 9 | ||
| 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 81 | 0 | 0 | ||
| 0 | 0 | 0 | 81 | 0 | ||
| 0 | 0 | 0 | 0 | 81 | ||
| 0 | 0 | 0 | 0 | 0 | ||
| 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 729 | 0 | ||
| 0 | 0 | 0 | 0 | 729 | ||
| 0 | 0 | 0 | 0 | 0 | ||
| 0 | 0 | 0 | 0 | 0 | ||
| 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6561 | ||
| 0 | 0 | 0 | 0 | 0 | ||
| 0 | 0 | 0 | 0 | 0 | ||
| 0 | 0 | 0 | 0 | 0 | ||
| 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | ||
| 0 | 0 | 0 | 0 | 0 | ||
| 0 | 0 | 0 | 0 | 0 | ||
| 0 | 0 | 0 | 0 | 0 | ||
| 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | ||
| 0 | 0 | 0 | 0 | 0 | ||
| 0 | 0 | 0 | 0 | 0 | ||
| 0 | 0 | 0 | 0 | 0 | ||
| 0 | 0 | 0 | 0 | 0 |
2.1.47 |
| 0 | 1 | 0 | 0 | 0 | ||
| 0 | 0 | 1 | 0 | 0 | ||
| 0 | 0 | 0 | 1 | 0 | ||
| 0 | 0 | 0 | 0 | 1 | ||
| 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | ||
| 0 | 0 | 0 | 1 | 0 | ||
| 0 | 0 | 0 | 0 | 1 | ||
| 0 | 0 | 0 | 0 | 0 | ||
| 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | ||
| 0 | 0 | 0 | 0 | 1 | ||
| 0 | 0 | 0 | 0 | 0 | ||
| 0 | 0 | 0 | 0 | 0 | ||
| 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | ||
| 0 | 0 | 0 | 0 | 0 | ||
| 0 | 0 | 0 | 0 | 0 | ||
| 0 | 0 | 0 | 0 | 0 | ||
| 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | ||
| 0 | 0 | 0 | 0 | 0 | ||
| 0 | 0 | 0 | 0 | 0 | ||
| 0 | 0 | 0 | 0 | 0 | ||
| 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | ||
| 0 | 0 | 0 | 0 | 0 | ||
| 0 | 0 | 0 | 0 | 0 | ||
| 0 | 0 | 0 | 0 | 0 | ||
| 0 | 0 | 0 | 0 | 0 |
2.2.1 |
| 7 | 5 |
| 8 | 4 |
| −13 | 512 | ||
| 23 | −712 |
2.2.1 |
| 3 | 9 |
| 7 | 2 |
| −257 | 319 | ||
| 757 | −119 |
2.2.1 |
| 4 | 8 |
| 9 | 2 |
| −132 | 18 | ||
| 964 | −116 |
2.2.8 |
7x1+2x2 | = | 6 | A−1= |
| ||||
−6x1−2x2 | = | −2 |
2.2.8 |
7x1+2x2 | = | −8 | A−1= |
| ||||
−6x1−2x2 | = | 4 |
2.2.9 |
| 1 | 2 |
| 8 | 18 |
| 1 |
| 16 |
| 5 |
| 34 |
| 2 |
| 12 |
| 6 |
| 40 |
| 9 | −1 | ||
| −4 | 12 |
| −7 | ||
| 4 |
| 11 | ||
| −3 |
| 6 | ||
| −2 |
| 14 | ||
| −4 |
| 1 | 0 | −7 | 11 | 6 | 14 | ||
| 0 | 1 | 4 | −3 | −2 | −4 |
2.2.9 |
| 1 | 2 |
| 8 | 18 |
| −1 |
| −2 |
| 5 |
| 34 |
| 3 |
| 22 |
| 4 |
| 22 |
| 9 | −1 | ||
| −4 | 12 |
| −7 | ||
| 3 |
| 11 | ||
| −3 |
| 5 | ||
| −1 |
| 14 | ||
| −5 |
| 1 | 0 | −7 | 11 | 5 | 14 | ||
| 0 | 1 | 3 | −3 | −1 | −5 |
2.2.9 |
| 1 | 2 |
| 7 | 16 |
| −2 |
| −8 |
| 1 |
| −3 |
| −2 |
| −20 |
| 5 |
| 27 |
| 8 | −1 | ||
| −72 | 12 |
| −8 | ||
| 3 |
| 11 | ||
| −5 |
| 4 | ||
| −3 |
| 13 | ||
| −4 |
| 1 | 0 | −8 | 11 | 4 | 13 | ||
| 0 | 1 | 3 | −5 | −3 | −4 |
2.2.10 |
2.2.10 |
2.2.11 |
2.2.11 |
2.2.12 |
2.2.13 |
2.2.13 |
2.2.14 |
2.2.14 |
2.2.14 |
2.2.14 |
2.2.14 |
2.2.15 |
| a | b | ||
| c | d |
| a | b | ||
| c | d |
| d | −b | ||
| −c | a |
2.2.15 |
| a | b | ||
| c | d |
| d | −b | ||
| −c | a |
| a | b | ||
| c | d |
2.2.16 |
| a | b | ||
| c | d |
| d | −b | ||
| −c | a |
| a | b | ||
| c | d |
2.2.16 |
| a | b | ||
| c | d |
| a | b | ||
| c | d |
| d | −b | ||
| −c | a |
2.2.17 |
2.2.17 |
2.2.17 |
2.2.17 |
2.2.18 |
2.2.18 |
2.2.39 |
| 1 | 6 |
| 4 | 23 |
| 1 | 6 | 1 | 0 | ||
| 4 | 23 | 0 | 1 |
| −23 | 6 | ||
| 4 | −1 |
2.2.39 |
| 1 | 3 |
| 4 | 11 |
| 1 | 3 | 1 | 0 | ||
| 4 | 11 | 0 | 1 |
| −11 | 3 | ||
| 4 | −1 |
2.2.40 |
| 4 | 3 |
| 3 | 5 |
| 4 | 3 | 1 | 0 | ||
| 3 | 5 | 0 | 1 |
| 511 | −311 | ||
| −311 | 411 |
2.2.40 |
| 6 | 3 |
| 4 | 7 |
| 6 | 3 | 1 | 0 | ||
| 4 | 7 | 0 | 1 |
| 730 | −110 | ||
| −215 | 15 |
2.2.41 |
| 1 | 0 | 2 |
| −3 | 1 | 3 |
| −4 | 4 | 3 |
| 1 | 0 | 2 | 1 | 0 | 0 | ||
| −3 | 1 | 3 | 0 | 1 | 0 | ||
| −4 | 4 | 3 | 0 | 0 | 1 |
| 925 | −825 | 225 | ||
| 325 | −1125 | 925 | ||
| 825 | 425 | −125 |
2.2.41 |
| 1 | 0 | −3 |
| −2 | 1 | −4 |
| 4 | 2 | −4 |
| 1 | 0 | −3 | 1 | 0 | 0 | ||
| −2 | 1 | −4 | 0 | 1 | 0 | ||
| 4 | 2 | −4 | 0 | 0 | 1 |
| 17 | −314 | 328 | ||
| −67 | 27 | 514 | ||
| −27 | −114 | 128 |
2.2.41 |
| 1 | 0 | −3 |
| −2 | 1 | 3 |
| 2 | −4 | 3 |
| 1 | 0 | −3 | 1 | 0 | 0 | ||
| −2 | 1 | 3 | 0 | 1 | 0 | ||
| 2 | −4 | 3 | 0 | 0 | 1 |
| −5 | −4 | −1 | ||
| −4 | −3 | −1 | ||
| −2 | −43 | −13 |
2.2.41 |
| 1 | 0 | 3 |
| −2 | 1 | 2 |
| −4 | 4 | 2 |
| 1 | 0 | 3 | 1 | 0 | 0 | ||
| −2 | 1 | 2 | 0 | 1 | 0 | ||
| −4 | 4 | 2 | 0 | 0 | 1 |
| 13 | −23 | 16 | ||
| 29 | −79 | 49 | ||
| 29 | 29 | −118 |
2.2.42 |
| 1 | −2 | 1 |
| 2 | −3 | 1 |
| −4 | 6 | −2 |
2.2.42 |
| 1 | −2 | 1 |
| −4 | 9 | −5 |
| 2 | −6 | 4 |
2.2.42 |
| 1 | −2 | 1 |
| 2 | −3 | 1 |
| 4 | −2 | −2 |
2.2.47 |
| 1 | 5 | ||
| 1 | 6 | ||
| 1 | 11 |
| 1 | 1 | −1 | ||
| −1 | 1 | 0 |
| −4 | 6 | −1 | ||
| −5 | 7 | −1 | ||
| −10 | 12 | −1 |
2.3.3 |
| 2 | 0 | 0 |
| −3 | −6 | 0 |
| 7 | 5 | −3 |
2.3.4 |
| 3 | 0 | −3 |
| 2 | 0 | 3 |
| −3 | 0 | 8 |
2.3.4 |
| 4 | 0 | −4 |
| 3 | 0 | 5 |
| −3 | 0 | 7 |
2.3.4 |
| 4 | 0 | −5 |
| 3 | 0 | 7 |
| −4 | 0 | 8 |
2.3.5 |
| 0 | 4 | −5 |
| 1 | 0 | 3 |
| −3 | −8 | 1 |
2.3.8 |
| 4 | 5 | 7 | 5 |
| 0 | 1 | 4 | 6 |
| 0 | 0 | 2 | 8 |
| 0 | 0 | 0 | 1 |
2.3.8 |
| 3 | 5 | 8 | 5 |
| 0 | 1 | 3 | 6 |
| 0 | 0 | 2 | 9 |
| 0 | 0 | 0 | 1 |
2.3.11 |
2.3.11 |
2.3.11 |
2.3.12 |
2.3.12 |
2.3.15 |
2.3.15 |
2.3.15 |
2.3.16 |
2.3.16 |
2.3.21 |
An m×n upper triangular matrix is one whose entries below the main diagonal are zeros, as is shown in the matrix to the right. When is a square upper triangular matrix invertible? Justify your answer. |
|
2.3.21 |
An m×n upper triangular matrix is one whose entries below the main diagonal are zeros, as is shown in the matrix to the right. When is a square upper triangular matrix invertible? Justify your answer. |
|
2.3.21 |
An m×n upper triangular matrix is one whose entries below the main diagonal are zeros, as is shown in the matrix to the right. When is a square upper triangular matrix invertible? Justify your answer. |
|
2.3.22 |
An m×n lower triangular matrix is one whose entries above the main diagonal are zeros, as is shown in the matrix to the right. When is a square lower triangular matrix invertible? Justify your answer. |
|
2.3.22 |
An m×n lower triangular matrix is one whose entries above the main diagonal are zeros, as is shown in the matrix to the right. When is a square lower triangular matrix invertible? Justify your answer. |
|
2.3.42 |
2.8.1 |
A set in ℝ2 is displayed to the right. Assume the set includes the bounding lines. Give a specific reason why the set H is not a subspace of ℝ2. (For instance, find two vectors in H whose sum is not in H, or find a vector in H with a scalar multiple that is not in H. Draw a picture.) |
2.8.1 |
A set in ℝ2 is displayed to the right. Assume the set includes the bounding lines. Give a specific reason why the set H is not a subspace of ℝ2. (For instance, find two vectors in H whose sum is not in H, or find a vector in H with a scalar multiple that is not in H. Draw a picture.) |
2.8.2 |
A set in ℝ2 is displayed to the right. Assume the set includes the bounding lines. Give a specific reason why the set H is not a subspace of ℝ2. (For instance, find two vectors in H whose sum is not in H, or find a vector in H with a scalar multiple that is not in H. Draw a picture.) |
2.8.2 |
A set in ℝ2 is displayed to the right. Assume the set includes the bounding lines. Give a specific reason why the set H is not a subspace of ℝ2. (For instance, find two vectors in H whose sum is not in H, or find a vector in H with a scalar multiple that is not in H. Draw a picture.) |
2.8.5 |
| 1 |
| 3 |
| −5 |
| −2 |
| −3 |
| 9 |
| −3 |
| −3 |
| 13 |
2.8.5 |
| 1 |
| 3 |
| −4 |
| −2 |
| −3 |
| 7 |
| −3 |
| −3 |
| 10 |
2.8.5 |
| 1 |
| 3 |
| −5 |
| −2 |
| −2 |
| 9 |
| −3 |
| −1 |
| 13 |
2.8.7 |
| 1 | ||
| −3 | ||
| 0 |
| 0 | ||
| 1 | ||
| 4 |
| 4 | ||
| −6 | ||
| 24 |
| 2 | ||
| −3 | ||
| 12 |
| v1 | v2 | v3 |
| A | p |
| A | p |
2.8.7 |
| 1 | ||
| −2 | ||
| 0 |
| 0 | ||
| 1 | ||
| 2 |
| 4 | ||
| −5 | ||
| 6 |
| 4 | ||
| −2 | ||
| 12 |
| v1 | v2 | v3 |
| A | p |
| A | p |
2.8.9 |
| −9 | ||
| 0 | ||
| 18 |
| −6 | ||
| 6 | ||
| 9 |
| 0 | ||
| −18 | ||
| 9 |
| 2 | ||
| 10 | ||
| −7 |
| v1 | v2 | v3 |
| A | p |
| A | p |
2.8.9 |
| −3 | ||
| 0 | ||
| 6 |
| −2 | ||
| 2 | ||
| 3 |
| 0 | ||
| −6 | ||
| 3 |
| 5 | ||
| 6 | ||
| −8 |
| v1 | v2 | v3 |
| A | p |
| A | p |
2.8.9 |
| −6 | ||
| 0 | ||
| 12 |
| −4 | ||
| 4 | ||
| 6 |
| 0 | ||
| −12 | ||
| 6 |
| 5 | ||
| 8 | ||
| −1 |
| v1 | v2 | v3 |
| A | p |
| A | p |
2.8.10 |
| −4 |
| 1 |
| 5 |
| −4 |
| 5 |
| 5 |
| 1 |
| −7 |
| 7 |
| −7 |
| 7 |
| 5 |
| v1 | v2 | v3 |
2.8.10 |
| −3 |
| 0 |
| 4 |
| −3 |
| 2 |
| 2 |
| 0 |
| −7 |
| 7 |
| −7 |
| 7 |
| 2 |
| v1 | v2 | v3 |
2.8.10 |
| −3 |
| 0 |
| 12 |
| −3 |
| 6 |
| 6 |
| 0 |
| −8 |
| 8 |
| −8 |
| 8 |
| 6 |
| v1 | v2 | v3 |
2.8.11 |
| 1 | 2 | 3 |
| 4 | 5 | 7 |
2.8.11 |
| −5 | 4 | 3 | −2 | 1 |
| 0 | 3 | −4 | 0 | 8 |
| 9 | 0 | 6 | −6 | 1 |
2.8.13 |
| 3 | 2 | −1 | −2 | ||
| −12 | −6 | 12 | 0 | ||
| 9 | 0 | −27 | 18 |
| 1 | ||
| 0 | ||
| 1 | ||
| 1 |
| 3 | ||
| −12 | ||
| 9 |
2.8.15 |
| −16 |
| 4 |
| −64 |
| −2 |
2.8.15 |
| −8 |
| 2 |
| −32 |
| 9 |
2.8.17 |
| 0 |
| 0 |
| −4 |
| 3 |
| 6 |
| 8 |
| 4 |
| 8 |
| 4 |
2.8.17 |
| 0 |
| 0 |
| −2 |
| 1 |
| 2 |
| 4 |
| 2 |
| 4 |
| 2 |
2.8.17 |
| 0 |
| 0 |
| −2 |
| −7 |
| 0 |
| 4 |
| 3 |
| 3 |
| 2 |
2.8.19 |
| 3 |
| −4 |
| 1 |
| 6 |
| 1 |
| −4 |
2.8.19 |
| 4 |
| −4 |
| 1 |
| 8 |
| 1 |
| −4 |
2.8.19 |
| 2 |
| −8 |
| 1 |
| 4 |
| 2 |
| −5 |
2.8.20 |
| 1 |
| 5 |
| −2 |
| 2 |
| −3 |
| −3 |
| 3 |
| −3 |
| 11 |
| 0 |
| 7 |
| −5 |
2.8.20 |
| 1 |
| 5 |
| −2 |
| 7 |
| −9 |
| 5 |
| 7 |
| −7 |
| −11 |
| 0 |
| 3 |
| −7 |
2.8.21 |
2.8.22 |
2.8.23 |
2.8.23 |
2.8.24 |
2.8.24 |
2.8.24 |
2.8.25 |
2.8.25 |
2.8.26 |
2.8.27 |
2.8.27 |
2.8.31 |
A | = |
| ~ |
|
| 5 | ||
| 5 | ||
| 2 |
| 8 | ||
| 2 | ||
| 1 |
| 3 | ||
| −4 | ||
| 1 | ||
| 0 |
| −5 | ||
| 8 | ||
| 0 | ||
| 1 |
2.8.31 |
A | = |
| ~ |
|
| 3 | ||
| 7 | ||
| 3 |
| 6 | ||
| 15 | ||
| 8 |
| 3 | ||
| −5 | ||
| 1 | ||
| 0 |
| −5 | ||
| 5 | ||
| 0 | ||
| 1 |
2.8.31 |
A | = |
| ~ |
|
| 3 | ||
| 6 | ||
| 3 |
| 7 | ||
| 10 | ||
| 7 |
| 3 | ||
| −4 | ||
| 1 | ||
| 0 |
| −6 | ||
| 8 | ||
| 0 | ||
| 1 |
2.8.33 |
A | = |
| ~ |
|
| 1 | ||
| −1 | ||
| −2 | ||
| 3 |
| 15 | ||
| 15 | ||
| −5 | ||
| 15 |
| −1 | ||
| 4 | ||
| 6 | ||
| −6 |
| 2 | ||
| −95 | ||
| 1 | ||
| 0 | ||
| 0 |
| −9 | ||
| 25 | ||
| 0 | ||
| −2 | ||
| 1 |
2.9.3 |
| 2 |
| −3 |
| −1 |
| 4 |
| 6 |
| −4 |
| 4 | ||
| 2 |
2.9.3 |
| 2 |
| −3 |
| −1 |
| 3 |
| 6 |
| −6 |
| 4 | ||
| 2 |
2.9.3 |
| 2 |
| −5 |
| −1 |
| 4 |
| 4 |
| −7 |
| 3 | ||
| 2 |
2.9.5 |
| 1 |
| 2 |
| −4 |
| −4 |
| −7 |
| 15 |
| −7 |
| −12 |
| 26 |
| 1 | ||
| 2 |
2.9.5 |
| 1 |
| 2 |
| −3 |
| −4 |
| −7 |
| 11 |
| −11 |
| −18 |
| 29 |
| 5 | ||
| 4 |
2.9.5 |
| 1 |
| 4 |
| −3 |
| −3 |
| −11 |
| 8 |
| −5 |
| −17 |
| 12 |
| 4 | ||
| 3 |
2.9.9 |
| 1 | 3 | 5 | −4 | ||
| 5 | 15 | 1 | 3 | ||
| 3 | 9 | −6 | 7 | ||
| 4 | 12 | 0 | 13 |
| 1 | 3 | 3 | 6 | ||
| 0 | 0 | 7 | −8 | ||
| 0 | 0 | 0 | 7 | ||
| 0 | 0 | 0 | 0 |
| 1 | ||
| 5 | ||
| 3 | ||
| 4 |
| 5 | ||
| 1 | ||
| −6 | ||
| 0 |
| −4 | ||
| 3 | ||
| 7 | ||
| 13 |
| −3 | ||
| 1 | ||
| 0 | ||
| 0 |
2.9.9 |
| 1 | 2 | 5 | −6 | ||
| 3 | 6 | 1 | 4 | ||
| 6 | 12 | −5 | 10 | ||
| 4 | 8 | 0 | 12 |
| 1 | 2 | 6 | 3 | ||
| 0 | 0 | 3 | −8 | ||
| 0 | 0 | 0 | 3 | ||
| 0 | 0 | 0 | 0 |
| 1 | ||
| 3 | ||
| 6 | ||
| 4 |
| 5 | ||
| 1 | ||
| −5 | ||
| 0 |
| −6 | ||
| 4 | ||
| 10 | ||
| 12 |
| −2 | ||
| 1 | ||
| 0 | ||
| 0 |
2.9.9 |
| 1 | 2 | 3 | −7 | ||
| 4 | 8 | 1 | 5 | ||
| 3 | 6 | −6 | 12 | ||
| 6 | 12 | 0 | 19 |
| 1 | 2 | 3 | 4 | ||
| 0 | 0 | 4 | −9 | ||
| 0 | 0 | 0 | 4 | ||
| 0 | 0 | 0 | 0 |
| 1 | ||
| 4 | ||
| 3 | ||
| 6 |
| 3 | ||
| 1 | ||
| −6 | ||
| 0 |
| −7 | ||
| 5 | ||
| 12 | ||
| 19 |
| −2 | ||
| 1 | ||
| 0 | ||
| 0 |
2.9.11 |
| 1 | 2 | −1 | −1 | −7 | ||
| 2 | 5 | 0 | 1 | −6 | ||
| −3 | −9 | −3 | −1 | 17 | ||
| 3 | 10 | 5 | 4 | −9 |
| 1 | 2 | −1 | −1 | −7 | ||
| 0 | 1 | 2 | 3 | 8 | ||
| 0 | 0 | 0 | 1 | 4 | ||
| 0 | 0 | 0 | 0 | 0 |
| 1 | ||
| 2 | ||
| −3 | ||
| 3 |
| 2 | ||
| 5 | ||
| −9 | ||
| 10 |
| −1 | ||
| 1 | ||
| −1 | ||
| 4 |
| 5 | ||
| −2 | ||
| 1 | ||
| 0 | ||
| 0 |
| −5 | ||
| 4 | ||
| 0 | ||
| −4 | ||
| 1 |
2.9.11 |
| 1 | 2 | 3 | 0 | −4 | ||
| 2 | 5 | 10 | 3 | 0 | ||
| −3 | −9 | −21 | −4 | 8 | ||
| 3 | 10 | 25 | 7 | 0 |
| 1 | 2 | 3 | 0 | −4 | ||
| 0 | 1 | 4 | 3 | 8 | ||
| 0 | 0 | 0 | 1 | 4 | ||
| 0 | 0 | 0 | 0 | 0 |
| 1 | ||
| 2 | ||
| −3 | ||
| 3 |
| 2 | ||
| 5 | ||
| −9 | ||
| 10 |
| 0 | ||
| 3 | ||
| −4 | ||
| 7 |
| 5 | ||
| −4 | ||
| 1 | ||
| 0 | ||
| 0 |
| −4 | ||
| 4 | ||
| 0 | ||
| −4 | ||
| 1 |
2.9.11 |
| 1 | 3 | 3 | 2 | 0 | ||
| 2 | 7 | 10 | 6 | 3 | ||
| −3 | −12 | −21 | −7 | 6 | ||
| 3 | 13 | 25 | 9 | −3 |
| 1 | 3 | 3 | 2 | 0 | ||
| 0 | 1 | 4 | 2 | 3 | ||
| 0 | 0 | 0 | 1 | 3 | ||
| 0 | 0 | 0 | 0 | 0 |
| 1 | ||
| 2 | ||
| −3 | ||
| 3 |
| 3 | ||
| 7 | ||
| −12 | ||
| 13 |
| 2 | ||
| 6 | ||
| −7 | ||
| 9 |
| 9 | ||
| −4 | ||
| 1 | ||
| 0 | ||
| 0 |
| −3 | ||
| 3 | ||
| 0 | ||
| −3 | ||
| 1 |
2.9.12 |
| 1 | 2 | −3 | 6 | 4 | ||
| 5 | 10 | −4 | −3 | 13 | ||
| 4 | 8 | −5 | 3 | 8 | ||
| −2 | −4 | 3 | −3 | −8 |
| 1 | 2 | −3 | 6 | 4 | ||
| 0 | 0 | 1 | −3 | 0 | ||
| 0 | 0 | 0 | 0 | −8 | ||
| 0 | 0 | 0 | 0 | 0 |
| 1 | ||
| 5 | ||
| 4 | ||
| −2 |
| −3 | ||
| −4 | ||
| −5 | ||
| 3 |
| 4 | ||
| 13 | ||
| 8 | ||
| −8 |
| −2 | ||
| 1 | ||
| 0 | ||
| 0 | ||
| 0 |
| 3 | ||
| 0 | ||
| 3 | ||
| 1 | ||
| 0 |
2.9.13 |
| 1 | ||
| −3 | ||
| 4 | ||
| −6 |
| −3 | ||
| 9 | ||
| −12 | ||
| 18 |
| 2 | ||
| −1 | ||
| 5 | ||
| 6 |
| −5 | ||
| 7 | ||
| −2 | ||
| 12 |
| 1 | ||
| −3 | ||
| 4 | ||
| −6 |
| 2 | ||
| −1 | ||
| 5 | ||
| 6 |
| −5 | ||
| 7 | ||
| −2 | ||
| 12 |
2.9.13 |
| 1 | ||
| −5 | ||
| 6 | ||
| −2 |
| −3 | ||
| 15 | ||
| −18 | ||
| 6 |
| 2 | ||
| −1 | ||
| 6 | ||
| 3 |
| −5 | ||
| 3 | ||
| −3 | ||
| 9 |
| 1 | ||
| −5 | ||
| 6 | ||
| −2 |
| 2 | ||
| −1 | ||
| 6 | ||
| 3 |
| −5 | ||
| 3 | ||
| −3 | ||
| 9 |
2.9.13 |
| 1 | ||
| −6 | ||
| 3 | ||
| −2 |
| −2 | ||
| 12 | ||
| −6 | ||
| 4 |
| 3 | ||
| −1 | ||
| 4 | ||
| 5 |
| −8 | ||
| 5 | ||
| −6 | ||
| 10 |
| 1 | ||
| −6 | ||
| 3 | ||
| −2 |
| 3 | ||
| −1 | ||
| 4 | ||
| 5 |
| −8 | ||
| 5 | ||
| −6 | ||
| 10 |
2.9.15 |
2.9.15 |
2.9.17 |
2.9.17 |
2.9.17 |
2.9.18 |
2.9.18 |
2.9.19 |
2.9.19 |
2.9.21 |
2.9.21 |
2.9.22 |
2.9.22 |
2.9.23 |
2.9.23 |
2.9.24 |
2.9.25 |
2.9.27 |
2.9.29 |
2.9.29 |
2.9.37 |
| 12 | ||
| −5 | ||
| 11 | ||
| 9 |
| 15 | ||
| −8 | ||
| 14 | ||
| 12 |
| 20 | ||
| −13 | ||
| 19 | ||
| 17 |
| v1 | v2 | x |
| 12 | 15 | 20 | ||
| −5 | −8 | −13 | ||
| 11 | 14 | 19 | ||
| 9 | 12 | 17 |
| 1 | 0 | −53 | ||
| 0 | 1 | 83 | ||
| 0 | 0 | 0 | ||
| 0 | 0 | 0 |
| B | x |
| x |
| −53 | ||
| 83 |
2.9.37 |
| 13 | ||
| −5 | ||
| 12 | ||
| 7 |
| 16 | ||
| −8 | ||
| 15 | ||
| 10 |
| 21 | ||
| −13 | ||
| 20 | ||
| 15 |
| v1 | v2 | x |
| 13 | 16 | 21 | ||
| −5 | −8 | −13 | ||
| 12 | 15 | 20 | ||
| 7 | 10 | 15 |
| 1 | 0 | −53 | ||
| 0 | 1 | 83 | ||
| 0 | 0 | 0 | ||
| 0 | 0 | 0 |
| B | x |
| x |
| −53 | ||
| 83 |
2.9.37 |
| 10 | ||
| −7 | ||
| 9 | ||
| 8 |
| 13 | ||
| −10 | ||
| 12 | ||
| 11 |
| 17 | ||
| −14 | ||
| 16 | ||
| 15 |
| v1 | v2 | x |
| 10 | 13 | 17 | ||
| −7 | −10 | −14 | ||
| 9 | 12 | 16 | ||
| 8 | 11 | 15 |
| 1 | 0 | −43 | ||
| 0 | 1 | 73 | ||
| 0 | 0 | 0 | ||
| 0 | 0 | 0 |
| B | x |
| x |
| −43 | ||
| 73 |
2.9.38 |
| −7 | ||
| 2 | ||
| −9 | ||
| 6 |
| 8 | ||
| −3 | ||
| 6 | ||
| −3 |
| −9 | ||
| 4 | ||
| −7 | ||
| −3 |
| −27 | ||
| 7 | ||
| −43 | ||
| 24 |
| v1 | v2 | v3 | x |
| −7 | 8 | −9 | −27 | ||
| 2 | −3 | 4 | 7 | ||
| −9 | 6 | −7 | −43 | ||
| 6 | −3 | −3 | 24 |
| 1 | 0 | 0 | 6 | ||
| 0 | 1 | 0 | 3 | ||
| 0 | 0 | 1 | 1 | ||
| 0 | 0 | 0 | 0 |
| B | x |
| B | x |
| x |
| 6 | ||
| 3 | ||
| 1 |
3.1.1 |
| 3 | 0 | 4 |
| 2 | 3 | 3 |
| 0 | 4 | −1 |
3.1.1 |
| 3 | 0 | 4 |
| 2 | 4 | 2 |
| 0 | 4 | −2 |
3.1.1 |
| 2 | 0 | 4 |
| 3 | 4 | 3 |
| 0 | 5 | −2 |
3.1.9 |
Compute the determinant by cofactor expansion. At each step, choose a row or column that involves the least amount of computation. |
|
| 9 | 0 | 0 | 5 |
| 8 | 7 | 3 | −3 |
| 3 | 0 | 0 | 0 |
| 7 | 3 | 1 | 1 |
3.1.9 |
Compute the determinant by cofactor expansion. At each step, choose a row or column that involves the least amount of computation. |
|
| 5 | 0 | 0 | 5 |
| 6 | 7 | 3 | −2 |
| 2 | 0 | 0 | 0 |
| 3 | 3 | 1 | 1 |
3.1.10 |
| 3 | −3 | 3 | 5 |
| 0 | 0 | 4 | 0 |
| 1 | −6 | −4 | 5 |
| 5 | 0 | 5 | 3 |
3.1.10 |
| 4 | −5 | 3 | 5 |
| 0 | 0 | 2 | 0 |
| 3 | −2 | −7 | 5 |
| 6 | 0 | 5 | 3 |
3.1.15 |
The expansion of a 3×3 determinant can be remembered by this device. Write a second copy of the first two columns to the right of the matrix, and compute the determinant by multiplying entries on six diagonals. Add the downward diagonal products and subtract the upward products. Use this method to compute the following determinant.
| − − − + + + a11 a12 a13 a11 a12 a21 a22 a23 a21 a22 a31 a32 a33 a31 a32 |
| −1 | −4 | 2 |
| 5 | 0 | −1 |
| 0 | −3 | 2 |
3.1.15 |
The expansion of a 3×3 determinant can be remembered by this device. Write a second copy of the first two columns to the right of the matrix, and compute the determinant by multiplying entries on six diagonals. Add the downward diagonal products and subtract the upward products. Use this method to compute the following determinant.
| − − − + + + a11 a12 a13 a11 a12 a21 a22 a23 a21 a22 a31 a32 a33 a31 a32 |
| −5 | −4 | −2 |
| 3 | 4 | 0 |
| 0 | −4 | −3 |
3.1.20 |
| a | b |
| c | d |
| a | b |
| 3c | 3d |
3.1.20 |
| a | b |
| c | d |
| c | d |
| a | b |
3.1.21 |
| 8 | 4 |
| 2 | 5 |
| 8 | 4 |
| 2+8k | 5+4k |
3.1.21 |
| 2 | 6 |
| 9 | 2 |
| 2 | 6 |
| 9+2k | 2+6k |
3.1.21 |
| 3 | 6 |
| 6 | 2 |
| 3 | 6 |
| 6+3k | 2+6k |
3.1.22 |
| a | b | ||
| c | d |
| a+kc | b+kd | ||
| c | d |
3.1.22 |
| a | b | ||
| c | d |
| a+kc | b+kd | ||
| c | d |
3.1.23 |
| 3 | 6 | 5 | ||
| 9 | 7 | 8 | ||
| a | b | c |
| a | b | c | ||
| 9 | 7 | 8 | ||
| 3 | 6 | 5 |
3.1.23 |
| 4 | 7 | 1 | ||
| a | b | c | ||
| 2 | 6 | 5 |
| a | b | c | ||
| 4 | 7 | 1 | ||
| 2 | 6 | 5 |
3.1.26 |
| 0 | 1 | 0 | ||
| 1 | 0 | 0 | ||
| 0 | 0 | 1 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
3.1.37 |
| 1 | 2 | ||
| 7 | 9 |
| 4 | 8 | ||
| 28 | 36 |
3.1.37 |
| 4 | 6 | ||
| 5 | 2 |
| 16 | 24 | ||
| 20 | 8 |
3.1.37 |
| 5 | 1 | ||
| 8 | 9 |
| 20 | 4 | ||
| 32 | 36 |
3.1.39 |
3.1.39 |
3.1.40 |
3.1.40 |
3.1.40 |
3.1.42 |
3.1.42 |
3.1.42 |
3.2.1 |
| −6 | 0 | 2 |
| −18 | −2 | −6 |
| −5 | −7 | 9 |
| −18 | −2 | −6 |
| −6 | 0 | 2 |
| −5 | −7 | 9 |
3.2.1 |
| 2 | −4 | −9 |
| −4 | 2 | 6 |
| 5 | 0 | −1 |
| −4 | 2 | 6 |
| 2 | −4 | −9 |
| 5 | 0 | −1 |
3.2.2 |
| 3 | −6 | 9 |
| 3 | 5 | −5 |
| 1 | 3 | 3 |
| 1 | −2 | 3 |
| 3 | 5 | −5 |
| 1 | 3 | 3 |
3.2.2 |
| 3 | −6 | 9 |
| 3 | 5 | −5 |
| 1 | 3 | 3 |
| 1 | −2 | 3 |
| 3 | 5 | −5 |
| 1 | 3 | 3 |
3.2.4 |
| 4 | 4 | 3 |
| 12 | −9 | 8 |
| −4 | 5 | −1 |
| 4 | 4 | 3 |
| 0 | −21 | −1 |
| −4 | 5 | −1 |
3.2.4 |
| −8 | −5 | −2 |
| −24 | 4 | 5 |
| 0 | −2 | −7 |
| −8 | −5 | −2 |
| 0 | 19 | 11 |
| 0 | −2 | −7 |
3.2.5 |
| 1 | 5 | −7 |
| 1 | 6 | 5 |
| −1 | −4 | 7 |
| 1 | 5 | −7 |
| 1 | 6 | 5 |
| −1 | −4 | 7 |
| 1 | 0 | 0 | ||
| 0 | 1 | 0 | ||
| 0 | 0 | 1 |
| 1 | 5 | −7 |
| 1 | 6 | 5 |
| −1 | −4 | 7 |
3.2.5 |
| 1 | 5 | −7 |
| −1 | −4 | −5 |
| 2 | 8 | 7 |
| 1 | 5 | −7 |
| −1 | −4 | −5 |
| 2 | 8 | 7 |
| 1 | 0 | 0 | ||
| 0 | 1 | 0 | ||
| 0 | 0 | 1 |
| 1 | 5 | −7 |
| −1 | −4 | −5 |
| 2 | 8 | 7 |
3.2.5 |
| 1 | 5 | −7 |
| 1 | 6 | 5 |
| −1 | −4 | 7 |
| 1 | 5 | −7 |
| 1 | 6 | 5 |
| −1 | −4 | 7 |
| 1 | 0 | 0 | ||
| 0 | 1 | 0 | ||
| 0 | 0 | 1 |
| 1 | 5 | −7 |
| 1 | 6 | 5 |
| −1 | −4 | 7 |
3.2.7 |
| 1 | −1 | −3 | 0 |
| 4 | −3 | 3 | 2 |
| 3 | −1 | −4 | 2 |
| −1 | 3 | 8 | 2 |
| 1 | −1 | −3 | 0 | ||
| 4 | −3 | 3 | 2 | ||
| 3 | −1 | −4 | 2 | ||
| −1 | 3 | 8 | 2 |
| 1 | 0 | 0 | 2625 | ||
| 0 | 1 | 0 | 45 | ||
| 0 | 0 | 1 | 225 | ||
| 0 | 0 | 0 | 0 |
| 1 | −1 | −3 | 0 |
| 4 | −3 | 3 | 2 |
| 3 | −1 | −4 | 2 |
| −1 | 3 | 8 | 2 |
3.2.15 |
| a | b | c |
| d | e | f |
| g | h | i |
| a | b | c |
| d | e | f |
| 4g | 4h | 4i |
| a | b | c |
| d | e | f |
| 4g | 4h | 4i |
3.2.15 |
| a | b | c |
| d | e | f |
| g | h | i |
| a | b | c |
| d | e | f |
| 5g | 5h | 5i |
| a | b | c |
| d | e | f |
| 5g | 5h | 5i |
3.2.20 |
| a | b | c |
| d | e | f |
| g | h | i |
| a | b | c |
| g | h | i |
| d | e | f |
3.2.20 |
| a | b | c |
| d | e | f |
| g | h | i |
| a | b | c |
| g | h | i |
| d | e | f |
3.2.20 |
| a | b | c |
| d | e | f |
| g | h | i |
| a | b | c |
| g | h | i |
| d | e | f |
3.2.21 |
| 5 | 0 | −1 |
| 1 | −3 | −2 |
| 0 | 5 | 3 |
3.2.22 |
| −20 | −5 | −1 |
| −4 | 15 | −2 |
| 0 | −25 | 3 |
3.2.22 |
| −10 | −3 | 2 |
| −2 | 9 | 4 |
| 0 | −15 | −6 |
3.2.22 |
| 20 | 1 | −3 |
| 4 | −3 | −6 |
| 0 | 5 | 9 |
3.2.25 |
| −3 | ||
| −3 | ||
| 1 |
| 2 | ||
| 4 | ||
| −3 |
| −3 | ||
| 0 | ||
| −4 |
3.2.25 |
| −2 | ||
| 7 | ||
| 3 |
| 1 | ||
| −6 | ||
| −2 |
| −2 | ||
| 0 | ||
| 6 |
3.2.25 |
| 2 | ||
| 2 | ||
| 6 |
| −3 | ||
| −1 | ||
| 2 |
| 2 | ||
| 0 | ||
| 1 |
3.2.28 |
| 2 | 6 | ||
| 1 | 3 |
| 2 | 3 | ||
| 2 | 3 |
| 1 | 2 | ||
| 0 | 0 |
3.2.28 |
| 2 | 6 | ||
| 1 | 3 |
| 2 | 3 | ||
| 2 | 3 |
| 1 | 2 | ||
| 0 | 0 |
3.2.28 |
| 2 | 3 | ||
| 2 | 3 |
| 1 | 2 | ||
| 0 | 0 |
| 2 | 6 | ||
| 1 | 3 |
3.2.29 |
3.2.29 |
3.2.29 |
3.2.32 |
| 2 | 3 | ||
| 0 | 4 |
3.2.32 |
| 2 | 3 | ||
| 0 | 4 |
3.2.33 |
| 2 | 0 | ||
| 1 | 0 |
| 3 | 0 | ||
| 5 | 0 |
| 1 | 0 | ||
| 0 | 1 |
| −1 | 0 | ||
| 0 | −1 |
3.2.35 |
| 1 | 0 | 2 |
| 1 | 2 | 3 |
| 2 | 3 | 1 |
3.2.35 |
| 1 | 0 | 2 |
| 2 | 1 | 2 |
| 1 | 2 | 1 |
3.2.35 |
| 2 | 0 | 1 |
| 2 | 1 | 3 |
| 1 | 2 | 1 |
3.2.38 |
3.2.38 |
3.2.51 |
3.2.51 |
3.3.1 |
3.3.1 |
3.3.1 |
3.3.3 |
−2x1+7x2= | 6 |
4x1+7x2= | 5 |
3.3.7 |
4sx1+7x2 | = | 3 |
8x1+3sx2 | = | −3 |
3.3.7 |
3sx1+4x2 | = | 5 |
9x1+4sx2 | = | −4 |
3.3.7 |
2sx1+4x2 | = | 5 |
9x1+3sx2 | = | −4 |
3.3.11 |
| 0 | −3 | −1 |
| 3 | 0 | 0 |
| −2 | 2 | 1 |
| 0 | 1 | 0 | ||
| −3 | −2 | −3 | ||
| 6 | 6 | 9 |
| 0 | 13 | 0 | ||
| −1 | −23 | −1 | ||
| 2 | 2 | 3 |
3.3.11 |
| 0 | −2 | −1 |
| 3 | 0 | 0 |
| −2 | 1 | 1 |
| 0 | 1 | 0 | ||
| −3 | −2 | −3 | ||
| 3 | 4 | 6 |
| 0 | 13 | 0 | ||
| −1 | −23 | −1 | ||
| 1 | 43 | 2 |
3.3.11 |
| 0 | −3 | −1 |
| 6 | 0 | 0 |
| −2 | 2 | 1 |
| 0 | 1 | 0 | ||
| −6 | −2 | −6 | ||
| 12 | 6 | 18 |
| 0 | 16 | 0 | ||
| −1 | −13 | −1 | ||
| 2 | 1 | 3 |
3.3.18 |
3.3.18 |
5.1.2 |
| 2 | −2 |
| −4 | 0 |
5.1.2 |
| −9 | 5 |
| −2 | −2 |
5.1.4 |
| 1 |
| 1 |
| −9 | 5 |
| −2 | −2 |
5.1.4 |
| −2 |
| 1 |
| −1 | 4 |
| 3 | 3 |
5.1.6 |
| 0 |
| 1 |
| 0 |
| 4 | 0 | 1 |
| −2 | 1 | 0 |
| −2 | 0 | 1 |
5.1.6 |
| −1 |
| 5 |
| 2 |
| −1 | 0 | −2 |
| 2 | 5 | −4 |
| 0 | 2 | −2 |
5.1.6 |
| 3 |
| −2 |
| 1 |
| −4 | 3 | 3 |
| 2 | −3 | −2 |
| −1 | 0 | −2 |
5.1.7 |
| 7 | 0 | −3 | ||
| 2 | 7 | 5 | ||
| −3 | 4 | 3 |
| 7 | 0 | −3 | ||
| 2 | 7 | 5 | ||
| −3 | 4 | 3 |
| −3 | ||
| −1 | ||
| 1 |
| 7 | 0 | −3 | ||
| 2 | 7 | 5 | ||
| −3 | 4 | 3 |
5.1.7 |
| 6 | 0 | −3 | ||
| 3 | 6 | 6 | ||
| −4 | 2 | 1 |
| 6 | 0 | −3 | ||
| 3 | 6 | 6 | ||
| −4 | 2 | 1 |
| −3 | ||
| −3 | ||
| 1 |
| 6 | 0 | −3 | ||
| 3 | 6 | 6 | ||
| −4 | 2 | 1 |
5.1.7 |
| 4 | 0 | −1 | ||
| 2 | 4 | 3 | ||
| −4 | 3 | −2 |
| 4 | 0 | −1 | ||
| 2 | 4 | 3 | ||
| −4 | 3 | −2 |
| −1 | ||
| 1 | ||
| 1 |
| 4 | 0 | −1 | ||
| 2 | 4 | 3 | ||
| −4 | 3 | −2 |
5.1.9 |
| 3 | 0 | ||
| −1 | 4 |
| 0 | ||
| 1 |
| 1 | ||
| 1 |
5.1.9 |
| 1 | 0 | ||
| −1 | 2 |
| 0 | ||
| 1 |
| 1 | ||
| 1 |
5.1.9 |
| 1 | 0 | ||
| −4 | 5 |
| 0 | ||
| 1 |
| 1 | ||
| 1 |
5.1.12 |
| 5 | 6 | ||
| −2 | −2 |
| −32 | ||
| 1 |
| −2 | ||
| 1 |
5.1.14 |
| 5 | 0 | 1 | ||
| −3 | 0 | 1 | ||
| −1 | −1 | 4 |
| −1 | ||
| 1 | ||
| 1 |
5.1.14 |
| 7 | 0 | −2 | ||
| 2 | 0 | −10 | ||
| −3 | 3 | 15 |
| 2 | ||
| −1 | ||
| 1 |
5.1.17 |
| 0 | 0 | 0 | ||
| 0 | 5 | −8 | ||
| 0 | 0 | 4 |
5.1.19 |
| 1 | 2 | 4 |
| 1 | 2 | 4 |
| 1 | 2 | 4 |
5.1.19 |
| 3 | 7 | 11 |
| 3 | 7 | 11 |
| 3 | 7 | 11 |
5.1.21 |
5.1.21 |
5.1.22 |
5.1.23 |
5.1.23 |
5.1.24 |
5.1.24 |
5.1.26 |
5.1.26 |
5.1.27 |
5.1.28 |
5.1.30 |
5.1.30 |
5.1.30 |
5.2.1 |
| 7 | 5 |
| 5 | 7 |
5.2.1 |
| 10 | 2 |
| 2 | 10 |
5.2.1 |
| 7 | 5 |
| 5 | 7 |
5.2.6 |
| 7 | −6 | ||
| 6 | 2 |
5.2.7 |
| −5 | −2 | ||
| 7 | −7 |
5.2.7 |
| 9 | 2 | ||
| −5 | 5 |
5.2.9 |
| 1 | 0 | 1 | ||
| −5 | 2 | −4 | ||
| 0 | 5 | 0 |
5.2.9 |
| 1 | 0 | −1 | ||
| 5 | 2 | −4 | ||
| 0 | 7 | 0 |
5.2.9 |
| 1 | 0 | −1 | ||
| 5 | 2 | −4 | ||
| 0 | 7 | 0 |
5.2.18 |
| 5 | −2 | 8 | −3 | ||
| 0 | 3 | h | 0 | ||
| 0 | 0 | 5 | 7 | ||
| 0 | 0 | 0 | 1 |
5.2.18 |
| 7 | −4 | 8 | 5 | ||
| 0 | 3 | h | 0 | ||
| 0 | 0 | 7 | 4 | ||
| 0 | 0 | 0 | 4 |
5.2.18 |
| 8 | −3 | 6 | 5 | ||
| 0 | 5 | h | 0 | ||
| 0 | 0 | 8 | 3 | ||
| 0 | 0 | 0 | 1 |
5.2.27 |
5.2.27 |
5.2.28 |
5.3.1 |
| 1 | 2 |
| 2 | 3 |
| 1 | 0 |
| 0 | 2 |
| 61 | −30 | ||
| 90 | −44 |
5.3.1 |
| 1 | 4 |
| 2 | 7 |
| 1 | 0 |
| 0 | 3 |
| 641 | −320 | ||
| 1120 | −559 |
5.3.1 |
| 1 | 4 |
| 2 | 7 |
| 1 | 0 |
| 0 | 3 |
| 641 | −320 | ||
| 1120 | −559 |
5.3.5 |
| 3 | 1 | 1 |
| 1 | 3 | 1 |
| 1 | 1 | 3 |
| 1 | 1 | 2 | ||
| 1 | 0 | −2 | ||
| 1 | −1 | 0 |
| 5 | 0 | 0 |
| 0 | 2 | 0 |
| 0 | 0 | 2 |
| 13 | 13 | 13 | ||
| 13 | 13 | −23 | ||
| 16 | −13 | 16 |
| 1 | ||
| 0 | ||
| −1 |
| 2 | ||
| −2 | ||
| 0 |
| 1 | ||
| 1 | ||
| 1 |
5.3.5 |
| 2 | 1 | 1 |
| 2 | 3 | 2 |
| 1 | 1 | 2 |
| 1 | 2 | 2 | ||
| 2 | 0 | −2 | ||
| 1 | −2 | 0 |
| 5 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 14 | 14 | 14 | ||
| 18 | 18 | −38 | ||
| 14 | −14 | 14 |
| 2 | ||
| 0 | ||
| −2 |
| 2 | ||
| −2 | ||
| 0 |
| 1 | ||
| 2 | ||
| 1 |
5.3.5 |
| 2 | 2 | 1 |
| 1 | 3 | 1 |
| 1 | 2 | 2 |
| 2 | 1 | 2 | ||
| 2 | 0 | −1 | ||
| 2 | −1 | 0 |
| 5 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 18 | 14 | 18 | ||
| 14 | 12 | −34 | ||
| 14 | −12 | 14 |
| 1 | ||
| 0 | ||
| −1 |
| 2 | ||
| −1 | ||
| 0 |
| 2 | ||
| 2 | ||
| 2 |
5.3.7 |
| 1 | 0 | ||
| 8 | −1 |
| 1 | 0 | ||
| 0 | 4 |
| 1 | 0 | ||
| 4 | 1 |
| 1 | 0 | ||
| 0 | −1 |
| 8 | 0 | ||
| 0 | −8 |
5.3.7 |
| 7 | 0 | ||
| 8 | −7 |
| 7 | 0 | ||
| 4 | 1 |
| 7 | 0 | ||
| 0 | −7 |
| 7 | 0 | ||
| 0 | 4 |
| 8 | 0 | ||
| 0 | −8 |
5.3.7 |
| 8 | 0 | ||
| 10 | −8 |
| 8 | 0 | ||
| 5 | 1 |
| 8 | 0 | ||
| 0 | −8 |
| 10 | 0 | ||
| 0 | −10 |
| 8 | 0 | ||
| 0 | 5 |
5.3.8 |
| 5 | 1 | ||
| 0 | 5 |
| 1 | 0 | ||
| 0 | −1 |
| 5 | 0 | ||
| 0 | 5 |
| 5 | 0 | ||
| 0 | 1 |
5.3.8 |
| 18 | 1 | ||
| 0 | 18 |
| 1 | 0 | ||
| 0 | −1 |
| 18 | 0 | ||
| 0 | 1 |
| 18 | 0 | ||
| 0 | 18 |
5.3.11 |
| 10 | 17 | −26 | ||
| 6 | 23 | −28 | ||
| 6 | 18 | −23 |
| 1 | 3 | 2 | ||
| 1 | 2 | 4 | ||
| 1 | 2 | 3 |
| 1 | 0 | 0 | ||
| 0 | 4 | 0 | ||
| 0 | 0 | 5 |
5.3.11 |
| −5 | 14 | −8 | ||
| −9 | 16 | −6 | ||
| −9 | 11 | −1 |
| 1 | 2 | 1 | ||
| 1 | 3 | 3 | ||
| 1 | 3 | 4 |
| 1 | 0 | 0 | ||
| 0 | 4 | 0 | ||
| 0 | 0 | 5 |
5.3.11 |
| 8 | 12 | −18 | ||
| 4 | 18 | −20 | ||
| 4 | 13 | −15 |
| 1 | 3 | 2 | ||
| 1 | 2 | 4 | ||
| 1 | 2 | 3 |
| 2 | 0 | 0 | ||
| 0 | 4 | 0 | ||
| 0 | 0 | 5 |
5.3.18 |
| 4 | 6 | −1 | ||
| −2 | −4 | 1 | ||
| −2 | −6 | 3 |
| 1 | 0 | 0 | ||
| 0 | 1 | 0 | ||
| 0 | 0 | 2 |
| −1 | −6 | 1 | ||
| 1 | 2 | 0 | ||
| 1 | 0 | 2 |
| −1 | 0 | 0 | ||
| 0 | 2 | 0 | ||
| 0 | 0 | 2 |
5.3.18 |
| 17 | 44 | −11 | ||
| −11 | −38 | 11 | ||
| −22 | −88 | 28 |
| −1 | −4 | 1 | ||
| 1 | 1 | 0 | ||
| 2 | 0 | 1 |
| −5 | 0 | 0 | ||
| 0 | 6 | 0 | ||
| 0 | 0 | 6 |
| 22 | 0 | 0 | ||
| 0 | 22 | 0 | ||
| 0 | 0 | 6 |
5.3.18 |
| −8 | −12 | 3 | ||
| 8 | 12 | −2 | ||
| 8 | 8 | 2 |
| −3 | 1 | −1 | ||
| 2 | 0 | 1 | ||
| 2 | 4 | 0 |
| −2 | 0 | 0 | ||
| 0 | 4 | 0 | ||
| 0 | 0 | 4 |
| −2 | 0 | 0 | ||
| 0 | −2 | 0 | ||
| 0 | 0 | 4 |
5.3.21 |
5.3.21 |
5.3.22 |
5.3.22 |
5.3.23 |
5.3.23 |
5.3.24 |
5.3.24 |
5.3.25 |
5.3.26 |
5.3.28 |
5.3.30 |
5.3.30 |
5.3.35 |
| 9 | 2 | ||
| −12 | −1 |
| 1 | 1 | ||
| −2 | −3 |
| 5 | 0 | ||
| 0 | 3 |
| 3 | 1 | ||
| −2 | −1 |
| 3 | 0 | ||
| 0 | 5 |
| 1 | 1 | ||
| −3 | −2 |
5.3.35 |
| 6 | 1 | ||
| −6 | 1 |
| 1 | 1 | ||
| −2 | −3 |
| 4 | 0 | ||
| 0 | 3 |
| 3 | 1 | ||
| −2 | −1 |
| 3 | 0 | ||
| 0 | 4 |
| 1 | 1 | ||
| −3 | −2 |
5.3.35 |
| 9 | 2 | ||
| −12 | −1 |
| 1 | 1 | ||
| −2 | −3 |
| 5 | 0 | ||
| 0 | 3 |
| 3 | 1 | ||
| −2 | −1 |
| 3 | 0 | ||
| 0 | 5 |
| 1 | 1 | ||
| −3 | −2 |
5.3.37 |
| 1 | 1 | ||
| 0 | 1 |
| 1 | 0 | ||
| 0 | 0 |
| 1 | 1 | ||
| 1 | 1 |
| 1 | 0 | ||
| 1 | 0 |
| 1 | 1 | ||
| 0 | 0 |
| 1 | 0 | ||
| 0 | 1 |
5.3.37 |
| 1 | 1 | ||
| 0 | 0 |
| 1 | 1 | ||
| 1 | 1 |
| 1 | 0 | ||
| 0 | 1 |
| 1 | 1 | ||
| 0 | 1 |
| 1 | 0 | ||
| 0 | 0 |
| 1 | 0 | ||
| 1 | 0 |
5.3.37 |
| 1 | 0 | ||
| 0 | 1 |
| 1 | 0 | ||
| 1 | 0 |
| 1 | 1 | ||
| 1 | 1 |
| 1 | 1 | ||
| 0 | 0 |
| 1 | 1 | ||
| 0 | 1 |
| 1 | 0 | ||
| 0 | 0 |
5.5.1 |
| 1 | −17 | ||
| 1 | 9 |
| 1 | −17 | ||
| 1 | 9 |
| −4+i | ||
| 1 |
| i | ||
| 1 |
| 1 | ||
| −4+i |
| −4−i | ||
| 1 |
| 1 | ||
| −4−i |
| −i | ||
| 1 |
| −4+i | ||
| 1 |
| −i | ||
| 1 |
| −4−i | ||
| 1 |
| i | ||
| 1 |
| 1 | ||
| −4−i |
| 1 | ||
| −4+i |
5.5.1 |
| 1 | −17 | ||
| 1 | 9 |
| 1 | −17 | ||
| 1 | 9 |
| −i | ||
| 1 |
| 1 | ||
| −4−i |
| −4+i | ||
| 1 |
| 1 | ||
| −4+i |
| −4−i | ||
| 1 |
| i | ||
| 1 |
| −i | ||
| 1 |
| 1 | ||
| −4−i |
| i | ||
| 1 |
| 1 | ||
| −4+i |
| −4+i | ||
| 1 |
| −4−i | ||
| 1 |
5.5.1 |
| 1 | −26 | ||
| 1 | 11 |
| 1 | −26 | ||
| 1 | 11 |
| −i | ||
| 1 |
| −5−i | ||
| 1 |
| −5+i | ||
| 1 |
| 1 | ||
| −5+i |
| 1 | ||
| −5−i |
| i | ||
| 1 |
| 1 | ||
| −5+i |
| −5−i | ||
| 1 |
| 1 | ||
| −5−i |
| i | ||
| 1 |
| −5+i | ||
| 1 |
| −i | ||
| 1 |
5.5.3 |
| 2 | 1 | ||
| −45 | 8 |
| 3−6i | ||
| 45 |
| 3−6i | ||
| 6 |
| 3−6i | ||
| 6 |
| 3+6i | ||
| 45 |
| 3−6i | ||
| 45 |
| 3+6i | ||
| 5 |
| 3+6i | ||
| 45 |
| 3+6i | ||
| 5 |
5.5.3 |
| 2 | 1 | ||
| −25 | 8 |
| 3−4i | ||
| 4 |
| 3+4i | ||
| 5 |
| 3−4i | ||
| 25 |
| 3−4i | ||
| 25 |
| 3+4i | ||
| 25 |
| 3+4i | ||
| 25 |
| 3−4i | ||
| 4 |
| 3+4i | ||
| 5 |
5.5.3 |
| 6 | 1 | ||
| −17 | 4 |
| −1+4i | ||
| 5 |
| −1+4i | ||
| 5 |
| −1−4i | ||
| 4 |
| −1−4i | ||
| 17 |
| −1+4i | ||
| 17 |
| −1−4i | ||
| 17 |
| −1+4i | ||
| 17 |
| −1−4i | ||
| 4 |
5.5.5 |
| 0 | 1 | ||
| −162 | 18 |
| 0 | 1 | ||
| −162 | 18 |
| 1 | ||
| 9+9i |
| 1 | ||
| 9+i |
| 9+i | ||
| 1 |
| 9i | ||
| 1 |
| 1 | ||
| 9i |
| 9+9i | ||
| 1 |
| 1 | ||
| 9−i |
| 9−i | ||
| 1 |
| −9i | ||
| 1 |
| 9−9i | ||
| 1 |
| 1 | ||
| 9−9i |
| 1 | ||
| −9i |
5.5.6 |
| 2 | −2 | ||
| 2 | 2 |
| 2 | −2 | ||
| 2 | 2 |
| i | ||
| 1 |
| −i | ||
| 1 |
5.5.6 |
| 5 | 4 | ||
| −4 | 5 |
| 5 | 4 | ||
| −4 | 5 |
| −i | ||
| 1 |
| i | ||
| 1 |
5.5.6 |
| 1 | 3 | ||
| −3 | 1 |
| 1 | 3 | ||
| −3 | 1 |
| −i | ||
| 1 |
| i | ||
| 1 |
5.5.7 |
| 43 | −4 |
| 4 | 43 |
5.5.7 |
| −3 | 1 |
| −1 | −3 |
5.5.7 |
| 33 | −3 |
| 3 | 33 |
5.5.9 |
List the eigenvalues of A. The transformation x↦Ax is the composition of a rotation and a scaling. Give the angle φ of the rotation, where −π<φ≤π, and give the scale factor r. | A=
|
5.5.9 |
List the eigenvalues of A. The transformation x↦Ax is the composition of a rotation and a scaling. Give the angle φ of the rotation, where −π<φ≤π, and give the scale factor r. | A=
|
5.5.9 |
List the eigenvalues of A. The transformation x↦Ax is the composition of a rotation and a scaling. Give the angle φ of the rotation, where −π<φ≤π, and give the scale factor r. | A=
|
5.5.10 |
List the eigenvalues of A. The transformation x↦Ax is the composition of a rotation and a scaling. Give the angle φ of the rotation, where −π<φ≤π, and give the scale factor r. | A=
|
5.5.10 |
List the eigenvalues of A. The transformation x↦Ax is the composition of a rotation and a scaling. Give the angle φ of the rotation, where −π<φ≤π, and give the scale factor r. | A=
|
5.5.10 |
List the eigenvalues of A. The transformation x↦Ax is the composition of a rotation and a scaling. Give the angle φ of the rotation, where −π<φ≤π, and give the scale factor r. | A=
|
5.5.11 |
List the eigenvalues of A. The transformation x↦Ax is the composition of a rotation and a scaling. Give the angle φ of the rotation, where −π<φ≤π, and give the scale factor r. | A=
|
5.5.12 |
List the eigenvalues of A. The transformation x↦Ax is the composition of a rotation and a scaling. Give the angle φ of the rotation, where −π<φ≤π, and give the scale factor r. | A=
|
5.5.13 |
| a | −b | ||
| b | a |
| 1 | −5 | ||
| 1 | 5 |
| −2−i | ||
| 1 |
| −2+i | ||
| 1 |
| −2 | −1 | ||
| 1 | 0 |
| 3 | −1 | ||
| 1 | 3 |
| a | −b | ||
| b | a |
5.5.13 |
| a | −b | ||
| b | a |
| 1 | −10 | ||
| 1 | 7 |
| −3−i | ||
| 1 |
| −3+i | ||
| 1 |
| −3 | −1 | ||
| 1 | 0 |
| 4 | −1 | ||
| 1 | 4 |
| a | −b | ||
| b | a |
5.5.13 |
| a | −b | ||
| b | a |
| 1 | −2 | ||
| 1 | 3 |
| −1−i | ||
| 1 |
| −1+i | ||
| 1 |
| −1 | −1 | ||
| 1 | 0 |
| 2 | −1 | ||
| 1 | 2 |
| a | −b | ||
| b | a |
5.5.16 |
| a | −b | ||
| b | a |
| 2 | −37 | ||
| 1 | 4 |
| 6+i | ||
| −i |
| 6−i | ||
| i |
| 6 | 1 | ||
| 0 | −1 |
| 3 | 6 | ||
| −6 | 3 |
| a | −b | ||
| b | a |
5.5.16 |
| a | −b | ||
| b | a |
| 5 | −5 | ||
| 1 | 7 |
| 2+i | ||
| −i |
| 2−i | ||
| i |
| 2 | 1 | ||
| 0 | −1 |
| 6 | 2 | ||
| −2 | 6 |
| a | −b | ||
| b | a |
5.5.16 |
| a | −b | ||
| b | a |
| 4 | −5 | ||
| 1 | 2 |
| 2−i | ||
| −i |
| 2+i | ||
| i |
| 2 | −1 | ||
| 0 | −1 |
| 3 | 2 | ||
| −2 | 3 |
| a | −b | ||
| b | a |
5.5.17 |
| a | −b | ||
| b | a |
| 1 | −0.75 | ||
| 2.4 | −1.4 |
| 12 | 14 | ||
| 1 | 0 |
| −15 | 35 | ||
| −35 | −15 |
| a | −b | ||
| b | a |
5.5.17 |
| a | −b | ||
| b | a |
| 1 | −2 | ||
| 0.8 | −1.4 |
| 32 | 12 | ||
| 1 | 0 |
| −15 | 25 | ||
| −25 | −15 |
| a | −b | ||
| b | a |
5.5.17 |
| a | −b | ||
| b | a |
| 0.8 | −0.6 | ||
| 3 | −1.6 |
| 25 | 15 | ||
| 1 | 0 |
| −25 | 35 | ||
| −35 | −25 |
| a | −b | ||
| b | a |
5.5.18 |
| a | −b | ||
| b | a |
| 0.6 | −0.08 | ||
| 1 | 0.2 |
| 15 | 15 | ||
| 1 | 0 |
| 25 | 15 | ||
| −15 | 25 |
| a | −b | ||
| b | a |
5.5.18 |
| a | −b | ||
| b | a |
| 1.2 | −0.8 | ||
| 1.6 | −0.4 |
| 12 | 12 | ||
| 1 | 0 |
| 25 | 45 | ||
| −45 | 25 |
| a | −b | ||
| b | a |
5.5.18 |
| a | −b | ||
| b | a |
| 1.2 | −0.8 | ||
| 1.6 | −0.4 |
| 12 | 12 | ||
| 1 | 0 |
| 25 | 45 | ||
| −45 | 25 |
| a | −b | ||
| b | a |
5.5.19 |
| a | −b | ||
| b | a |
| 1.52 | −0.7 | ||
| 0.56 | 0.4 |
| 1 | 12 | ||
| 1 | 0 |
| 2425 | 725 | ||
| −725 | 2425 |
| a | −b | ||
| b | a |
5.5.19 |
| a | −b | ||
| b | a |
| 1.2 | −0.5 | ||
| 1.04 | 0.4 |
| 513 | 1526 | ||
| 1 | 0 |
| 45 | 35 | ||
| −35 | 45 |
| a | −b | ||
| b | a |
5.5.19 |
| a | −b | ||
| b | a |
| 1 | −0.5 | ||
| 0.8 | 0.6 |
| 14 | 34 | ||
| 1 | 0 |
| 45 | 35 | ||
| −35 | 45 |
| a | −b | ||
| b | a |
6.1.1 |
| −2 | ||
| 5 |
| 7 | ||
| 6 |
6.1.1 |
| −1 | ||
| 4 |
| 5 | ||
| 3 |
6.1.1 |
| −2 | ||
| 3 |
| 6 | ||
| 7 |
6.1.2 |
| 3 | ||
| −4 | ||
| −5 |
| 6 | ||
| −3 | ||
| 3 |
6.1.2 |
| 3 | ||
| −4 | ||
| −5 |
| 7 | ||
| −3 | ||
| 2 |
6.1.4 |
| −4 | ||
| 5 |
| −441 | ||
| 541 |
6.1.4 |
| −2 | ||
| 3 |
| −213 | ||
| 313 |
6.1.7 |
| 5 | ||
| 1 | ||
| −5 |
6.1.7 |
| 2 | ||
| 2 | ||
| −7 |
6.1.9 |
| 72 | ||
| −320 |
| 941 | ||
| −4041 |
6.1.9 |
| 77 | ||
| −264 |
| 77275 | ||
| −264275 |
6.1.10 |
| −2 | ||
| −18 | ||
| −2 |
| −183 | ||
| −983 | ||
| −183 |
6.1.10 |
| −3 | ||
| −18 | ||
| 3 |
| −138 | ||
| −638 | ||
| 138 |
6.1.13 |
| 12 | ||
| −3 |
| −4 | ||
| −9 |
6.1.13 |
| 10 | ||
| −4 |
| −1 | ||
| −9 |
6.1.13 |
| 11 | ||
| −3 |
| −1 | ||
| −9 |
6.1.14 |
| 0 | ||
| −6 | ||
| 3 |
| −2 | ||
| −1 | ||
| 6 |
6.1.14 |
| 0 | ||
| −6 | ||
| 2 |
| −3 | ||
| −1 | ||
| 5 |
6.1.15 |
| 8 | ||
| −3 |
| −3 | ||
| −2 |
6.1.15 |
| 8 | ||
| −3 |
| −2 | ||
| −2 |
6.1.16 |
| 10 | ||
| 4 | ||
| 2 |
| 1 | ||
| −4 | ||
| 3 |
6.1.16 |
| 10 | ||
| 5 | ||
| 15 |
| 1 | ||
| −5 | ||
| 1 |
6.1.18 |
| −4 |
| 4 |
| 9 |
| 0 |
| 1 |
| −8 |
| 16 |
| −4 |
6.1.19 |
6.1.19 |
6.1.22 |
6.1.22 |
6.1.22 |
6.1.23 |
6.1.23 |
6.1.24 |
6.1.24 |
6.1.24 |
6.1.27 |
6.1.27 |
6.2.2 |
| −2 |
| 1 |
| 1 |
| 1 |
| 2 |
| 0 |
| −2 |
| 1 |
| −5 |
| −2 |
| 1 |
| 1 |
| 1 |
| 2 |
| 0 |
| 1 |
| 2 |
| 0 |
| −2 |
| 1 |
| −5 |
6.2.2 |
| −4 |
| 1 |
| 1 |
| 1 |
| 4 |
| 0 |
| −4 |
| 1 |
| −17 |
| −4 |
| 1 |
| 1 |
| 1 |
| 4 |
| 0 |
| 1 |
| 4 |
| 0 |
| −4 |
| 1 |
| −17 |
6.2.2 |
| −2 |
| 1 |
| 1 |
| 1 |
| 2 |
| 0 |
| −2 |
| 1 |
| −5 |
| −2 |
| 1 |
| 1 |
| 1 |
| 2 |
| 0 |
| 1 |
| 2 |
| 0 |
| −2 |
| 1 |
| −5 |
6.2.3 |
| 2 | ||
| −5 | ||
| −1 |
| −8 | ||
| −4 | ||
| 4 |
| 3 | ||
| 1 | ||
| 1 |
6.2.3 |
| 1 | ||
| 3 | ||
| −1 |
| 0 | ||
| −1 | ||
| −3 |
| −10 | ||
| 3 | ||
| −1 |
6.2.4 |
| 4 |
| −5 |
| −7 |
| 0 |
| 0 |
| 0 |
| 2 |
| −4 |
| 4 |
6.2.4 |
| 0 |
| 0 |
| 0 |
| 2 |
| −3 |
| −7 |
| 4 |
| −2 |
| 2 |
6.2.7 |
| 2 | ||
| −4 |
| 8 | ||
| 4 |
| 6 | ||
| −7 |
6.2.7 |
| 2 | ||
| −6 |
| 12 | ||
| 4 |
| 2 | ||
| −2 |
6.2.10 |
| 4 | ||
| −4 | ||
| 0 |
| 2 | ||
| 2 | ||
| −1 |
| 1 | ||
| 1 | ||
| 4 |
| 4 | ||
| −3 | ||
| 1 |
6.2.10 |
| 3 | ||
| −3 | ||
| 0 |
| 2 | ||
| 2 | ||
| −1 |
| 1 | ||
| 1 | ||
| 4 |
| 5 | ||
| −2 | ||
| 1 |
6.2.10 |
| 4 | ||
| −4 | ||
| 0 |
| 3 | ||
| 3 | ||
| −1 |
| 1 | ||
| 1 | ||
| 6 |
| 4 | ||
| −3 | ||
| 1 |
6.2.11 |
| 1 | ||
| 7 |
| −3 | ||
| 9 |
| −2 | ||
| 6 |
6.2.11 |
| 8 | ||
| 9 |
| −3 | ||
| 6 |
| −2 | ||
| 4 |
6.2.13 |
| 1 |
| 6 |
| 5 |
| −7 |
| −52 | ||
| 72 |
| 72 | ||
| 52 |
6.2.13 |
| 1 |
| 4 |
| 3 |
| −5 |
| −32 | ||
| 52 |
| 52 | ||
| 32 |
6.2.15 |
| 4 | ||
| 7 |
| 6 | ||
| 8 |
6.2.15 |
| 9 | ||
| 6 |
| 8 | ||
| 6 |
6.2.15 |
| 4 | ||
| 2 |
| 8 | ||
| 6 |
6.2.17 |
| 23 | ||
| −13 | ||
| 13 |
| −13 | ||
| 0 | ||
| 23 |
| 26 | ||
| −16 | ||
| 16 |
| −15 | ||
| 0 | ||
| 25 |
6.2.17 |
| 23 | ||
| −13 | ||
| 13 |
| −15 | ||
| 0 | ||
| 25 |
| 23 | ||
| −16 | ||
| 16 |
| −55 | ||
| 0 | ||
| 255 |
6.2.19 |
| 0.6 | ||
| −0.8 |
| −0.8 | ||
| −0.6 |
6.2.19 |
| −0.8 | ||
| 0.6 |
| 0.6 | ||
| 0.8 |
6.2.23 |
| 0 | ||
| 1 |
| 1 | ||
| 1 |
| 1 | ||
| −1 |
| 1 | ||
| 1 |
6.2.23 |
| 0 | ||
| 1 |
| 1 | ||
| 1 |
| 1 | ||
| −1 |
| 1 | ||
| 1 |
6.2.23 |
| 0 | ||
| 1 |
| 1 | ||
| 1 |
| 1 | ||
| −1 |
| 1 | ||
| 1 |
6.2.24 |
6.2.24 |
6.2.25 |
6.2.25 |
6.2.26 |
6.2.26 |
6.2.28 |
6.2.28 |
6.2.29 |
6.2.29 |
6.2.32 |
6.3.4 |
| 4 |
| 5 |
| −4 |
| 4 |
| 5 |
| 0 |
| −5 |
| 4 |
| 0 |
| 4 |
| 5 |
| 0 |
6.3.4 |
| 5 |
| 3 |
| −2 |
| 4 |
| 5 |
| 0 |
| −5 |
| 4 |
| 0 |
| 5 |
| 3 |
| 0 |
6.3.4 |
| 3 |
| 5 |
| −1 |
| 3 |
| 6 |
| 0 |
| −6 |
| 3 |
| 0 |
| 3 |
| 5 |
| 0 |
6.3.8 |
| −6 | ||
| 4 | ||
| 7 |
| 1 | ||
| 1 | ||
| 2 |
| −3 | ||
| 9 | ||
| −3 |
| 1 | ||
| 5 | ||
| 3 |
| −7 | ||
| −1 | ||
| 4 |
6.3.8 |
| −2 | ||
| 3 | ||
| 7 |
| 1 | ||
| 1 | ||
| 3 |
| −2 | ||
| 5 | ||
| −1 |
| 65 | ||
| 4 | ||
| 285 |
| −165 | ||
| −1 | ||
| 75 |
6.3.8 |
| −7 | ||
| 2 | ||
| 9 |
| 1 | ||
| 1 | ||
| 3 |
| −2 | ||
| 5 | ||
| −1 |
| 1 | ||
| 92 | ||
| 112 |
| −8 | ||
| −52 | ||
| 72 |
6.3.9 |
| 4 | ||
| 2 | ||
| 2 | ||
| −1 |
| 1 | ||
| 1 | ||
| 0 | ||
| 1 |
| −1 | ||
| 5 | ||
| 3 | ||
| −4 |
| −1 | ||
| 0 | ||
| 1 | ||
| 1 |
| 4017 | ||
| 5517 | ||
| −117 | ||
| −1017 |
| 2817 | ||
| −2117 | ||
| 3517 | ||
| −717 |
6.3.9 |
| 4 | ||
| 6 | ||
| 6 | ||
| −1 |
| 1 | ||
| 1 | ||
| 0 | ||
| 1 |
| −1 | ||
| 5 | ||
| 3 | ||
| −4 |
| −1 | ||
| 0 | ||
| 1 | ||
| 1 |
| 8851 | ||
| 13117 | ||
| 16151 | ||
| −2251 |
| 11651 | ||
| −2917 | ||
| 14551 | ||
| −2951 |
6.3.12 |
| 3 | ||
| −1 | ||
| 1 | ||
| 19 |
| 1 | ||
| −3 | ||
| −1 | ||
| 1 |
| −1 | ||
| 1 | ||
| 0 | ||
| 4 |
| −2 | ||
| −2 | ||
| −2 | ||
| 18 |
6.3.12 |
| 7 | ||
| −1 | ||
| 1 | ||
| 14 |
| 1 | ||
| −1 | ||
| −1 | ||
| 2 |
| −2 | ||
| 2 | ||
| 0 | ||
| 2 |
| 3 | ||
| −3 | ||
| −5 | ||
| 12 |
6.3.13 |
| 3 | ||
| −6 | ||
| 2 | ||
| 3 |
| 2 | ||
| 1 | ||
| 2 | ||
| −1 |
| 1 | ||
| −1 | ||
| 0 | ||
| 1 |
| 215 | ||
| −3910 | ||
| 15 | ||
| 3910 |
6.3.13 |
| 3 | ||
| −6 | ||
| 2 | ||
| 3 |
| 3 | ||
| −1 | ||
| 2 | ||
| 2 |
| 1 | ||
| 1 | ||
| 0 | ||
| −1 |
| 136 | ||
| −6118 | ||
| 259 | ||
| 439 |
6.3.13 |
| 4 | ||
| −5 | ||
| 2 | ||
| 4 |
| 4 | ||
| 2 | ||
| 3 | ||
| −2 |
| 1 | ||
| −1 | ||
| 0 | ||
| 1 |
| 5311 | ||
| −4511 | ||
| 411 | ||
| 4511 |
6.3.15 |
| 6 |
| −8 |
| 5 |
| −2 |
| −5 |
| 1 |
| −3 |
| 1 |
| −1 |
6.3.15 |
| 5 |
| −8 |
| 5 |
| −3 |
| −5 |
| 1 |
| −4 |
| 2 |
| −2 |
6.3.16 |
| −3 |
| −1 |
| 1 |
| −13 |
| 1 |
| 2 |
| −1 |
| −2 |
| 4 |
| 1 |
| 0 |
| 3 |
| −6 |
| 2 |
| −2 |
| −10 |
6.3.16 |
| −3 |
| −1 |
| 1 |
| −13 |
| 1 |
| 2 |
| −1 |
| −2 |
| 4 |
| 1 |
| 0 |
| 3 |
| −6 |
| 2 |
| −2 |
| −10 |
6.3.17 |
| 3 | ||
| 9 | ||
| 1 |
| 23 | ||
| −13 | ||
| −23 |
| 23 | ||
| 23 | ||
| 13 |
| u1 | u2 |
| 1 | 0 | ||
| 0 | 1 |
| 89 | 29 | −29 | ||
| 29 | 59 | 49 | ||
| −29 | 49 | 59 |
| 409 | ||
| 559 | ||
| 359 |
| 409 | ||
| 559 | ||
| 359 |
6.3.17 |
| 4 | ||
| 8 | ||
| 1 |
| −23 | ||
| 23 | ||
| 13 |
| 23 | ||
| 13 | ||
| 23 |
| u1 | u2 |
| 1 | 0 | ||
| 0 | 1 |
| 89 | −29 | 29 | ||
| −29 | 59 | 49 | ||
| 29 | 49 | 59 |
| 2 | ||
| 4 | ||
| 5 |
| 2 | ||
| 4 | ||
| 5 |
6.3.19 |
| −2 | ||
| −2 | ||
| 2 |
| −2 | ||
| 1 | ||
| −1 |
| 0 | ||
| 0 | ||
| 1 |
| 0 | ||
| 12 | ||
| 12 |
6.3.19 |
| −1 | ||
| 1 | ||
| −1 |
| −2 | ||
| −1 | ||
| 1 |
| 0 | ||
| 0 | ||
| 1 |
| 0 | ||
| 12 | ||
| 12 |
6.3.19 |
| 2 | ||
| 1 | ||
| 1 |
| −3 | ||
| 3 | ||
| 3 |
| 0 | ||
| 0 | ||
| 1 |
| 0 | ||
| −12 | ||
| 12 |
6.3.21 |
6.3.21 |
6.3.22 |
6.3.22 |
6.3.23 |
6.3.24 |
6.3.24 |
6.3.25 |
6.3.25 |
6.4.1 |
The given set is a basis for a subspace W. Use the Gram-Schmidt process to produce an orthogonal basis for W. |
|
| 0 | ||
| 8 | ||
| 8 |
| 5 | ||
| 4 | ||
| −4 |
6.4.1 |
The given set is a basis for a subspace W. Use the Gram-Schmidt process to produce an orthogonal basis for W. |
|
| 0 | ||
| 8 | ||
| 8 |
| 8 | ||
| 6 | ||
| −6 |
6.4.10 |
Find an orthogonal basis for the column space of the matrix to the right. |
|
| −1 | ||
| 3 | ||
| 2 | ||
| 1 |
| 3 | ||
| 0 | ||
| 3 | ||
| −3 |
| 2 | ||
| 0 | ||
| 0 | ||
| 2 |
6.4.10 |
Find an orthogonal basis for the column space of the matrix to the right. |
|
| −1 | ||
| 2 | ||
| 1 | ||
| 1 |
| 2 | ||
| 0 | ||
| 2 | ||
| 0 |
| 0 | ||
| 2 | ||
| 0 | ||
| −4 |
6.4.10 |
Find an orthogonal basis for the column space of the matrix to the right. |
|
| −1 | ||
| 1 | ||
| 1 | ||
| 1 |
| 1 | ||
| −3 | ||
| 3 | ||
| 1 |
| 6 | ||
| 5 | ||
| 4 | ||
| −3 |
6.4.14 |
The columns of Q were obtained by applying the Gram-Schmidt process to the columns of A. Find an upper triangular matrix R such that A=QR. | A=
|
| 7 | 7 | ||
| 0 | 14 |
6.4.14 |
The columns of Q were obtained by applying the Gram-Schmidt process to the columns of A. Find an upper triangular matrix R such that A=QR. | A=
|
| 7 | 7 | ||
| 0 | 29 |
6.4.15 |
An orthogonal basis for the column space of matrix A is v1, v2, v3. Use this orthogonal basis to find a QR factorization of matrix A. | A=
|
| 12 | −232 | 219 | ||
| −12 | 232 | 319 | ||
| 0 | 432 | −119 | ||
| 12 | 232 | −119 | ||
| 12 | 232 | 219 |
| 2 | 8 | 7 | ||
| 0 | 42 | 32 | ||
| 0 | 0 | 19 |
6.4.17 |
6.4.17 |
6.4.21 |
6.4.21 |
6.5.1 |
| −1 | 2 | ||
| 2 | −3 | ||
| −1 | 1 |
| 2 | ||
| 1 | ||
| 3 |
| 6 | −9 | ||
| −9 | 14 |
| x1 | ||
| x2 |
| −3 | ||
| 4 |
| −2 | ||
| −1 |
6.5.1 |
| −1 | 1 | ||
| 1 | −2 | ||
| −1 | 1 |
| 1 | ||
| 2 | ||
| 3 |
| 3 | −4 | ||
| −4 | 6 |
| x1 | ||
| x2 |
| −2 | ||
| 0 |
| −6 | ||
| −4 |
6.5.3 |
| 1 | −4 | ||
| −1 | 4 | ||
| 0 | 2 | ||
| 4 | 7 |
| 5 | ||
| 1 | ||
| −3 | ||
| 4 |
| 18 | 20 | ||
| 20 | 85 |
| 20 | ||
| 6 |
| 158113 | ||
| −146565 |
6.5.4 |
| 1 | 1 | ||
| 1 | −1 | ||
| 1 | 1 |
| 4 | ||
| 6 | ||
| 0 |
| 3 | 1 | ||
| 1 | 3 |
| x1 | ||
| x2 |
| 10 | ||
| −2 |
| 4 | ||
| −2 |
6.5.5 |
| 1 | 1 | 0 | ||
| 1 | 0 | 1 | ||
| 1 | 0 | 1 | ||
| 1 | 1 | 0 |
| 3 | ||
| 11 | ||
| 3 | ||
| 1 |
| 7 | ||
| −5 | ||
| 0 |
| −1 | ||
| 1 | ||
| 1 |
6.5.5 |
| 1 | 0 | 1 | ||
| 1 | 0 | 1 | ||
| 1 | 1 | 0 | ||
| 1 | 1 | 0 |
| 2 | ||
| 10 | ||
| 1 | ||
| 3 |
| 6 | ||
| −4 | ||
| 0 |
| −1 | ||
| 1 | ||
| 1 |
6.5.7 |
| 1 | −2 | ||
| −1 | 2 | ||
| 0 | 4 | ||
| 5 | 9 |
| 4 | ||
| 1 | ||
| −2 | ||
| 4 |
| 1,5131,154 | ||
| −3491,154 |
6.5.8 |
| 1 | 2 |
| 1 | −1 |
| 1 | 1 |
| 3 |
| 8 |
| 0 |
| 5 |
| −2 |
6.5.10 |
| −1 | 4 |
| 1 | 8 |
| −1 | 4 |
| 11 |
| −1 |
| 3 |
| 7 | ||
| −1 | ||
| 7 |
| −5 | ||
| 12 |
6.5.13 |
| 9 | 7 | ||
| −5 | 1 | ||
| 9 | 7 |
| 10 | ||
| −7 | ||
| 6 |
| 9 | ||
| −5 |
| 9 | ||
| −3 |
| 46 | ||
| −50 | ||
| 46 |
| 60 | ||
| −48 | ||
| 60 |
6.5.13 |
| 8 | 9 | ||
| −6 | 1 | ||
| 8 | 9 |
| 11 | ||
| −5 | ||
| 9 |
| 5 | ||
| −3 |
| 5 | ||
| −6 |
| 13 | ||
| −33 | ||
| 13 |
| −14 | ||
| −36 | ||
| −14 |
6.5.15 |
| 2 | 3 | ||
| 2 | 4 | ||
| 1 | 1 |
| 23 | −13 | ||
| 23 | 23 | ||
| 13 | −23 |
| 3 | 5 | ||
| 0 | 1 |
| 7 | ||
| 3 | ||
| 1 |
| 4 | ||
| −1 |
6.5.17 |
6.5.18 |
6.5.19 |
6.5.20 |
6.5.21 |
6.5.22 |
6.6.1 |
6.6.3 |
6.6.9 |
6.6.13 |
| 1 | 1 | ||
| 2 | 4 | ||
| 3 | 9 | ||
| 4 | 16 | ||
| 5 | 25 |
| 1 | 1 | 1 | 1 | 1 | ||
| 1 | 2 | 3 | 4 | 5 | ||
| 1 | 4 | 9 | 16 | 25 |
| 1 | 1 | 1 | ||
| 1 | 2 | 4 | ||
| 1 | 3 | 9 | ||
| 1 | 4 | 16 | ||
| 1 | 5 | 25 |
| 1 | 2 | 3 | 4 | 5 | ||
| 1 | 4 | 9 | 16 | 25 |
| 1.7 | ||
| 2.8 | ||
| 3.3 | ||
| 3.6 | ||
| 3.9 |
| β1 | ||
| β2 | ||
| β3 | ||
| β4 | ||
| β5 |
| 0 |
| β1 |
| β2 |
| x |
| x2 |
| β1 | ||
| β2 |
*9.1.B1 |
*9.1.B1 |
*9.1.B1 |
*9.1.B2 |
*9.1.B2 |
*9.1.B2 |
*9.1.B3 |
(a) | 17i | (b) | 3 |
(c) | 11−12i | (d) | 14+13i |
*9.1.B3 |
(a) | 18i | (b) | 10 |
(c) | 6−4i | (d) | 3+5i |
*9.1.B3 |
(a) | 19i | (b) | 4 |
(c) | 16−11i | (d) | 15+5i |
*9.1.B5 |
(a) | 34−8i | (b) | 1−6i4+8i |
*9.1.B5 |
(a) | 67−8i | (b) | 5−9i7+8i |
*9.1.B5 |
(a) | 95−8i | (b) | 4−6i5+8i |
*9.1.B6 |
*9.1.B6 |
*9.1.B6 |
*9.1.B7 |
*9.1.B7 |
*9.1.B7 |
*9.2.B1 |
(a) | z2−iz3 | = | 5 |
iz1−z2+(−1−i)z3 | = | −7−9i | |
−2iz1+z2+(3+2i)z3 | = | 15+16i |
| 1+2i | ||
| 3+4i | ||
| 4+2i |
| 1+2i | ||
| 0 | ||
| 0 |
| 3+4i | ||
| 4+2i | ||
| 1 |
| 1+2i | ||
| 3+4i | ||
| 0 |
| 4+2i | ||
| 0 | ||
| 1 |
(b) | z1+(−1−i)z2+iz3 | = | −2−i |
iz1+(1−i)z2+3iz4 | = | i | |
z1+(−1−i)z2+(1+i)z3+3iz4 | = | −3+2i |
| 1 | ||
| 0 | ||
| −1+3i | ||
| 0 |
| 1 | ||
| 0 | ||
| −1+3i | ||
| 0 |
| 1+i | ||
| 1 | ||
| 0 | ||
| 0 |
| −3 | ||
| 0 | ||
| −3i | ||
| 1 |
| 1 | ||
| 0 | ||
| −1+3i | ||
| 0 |
| 1+i | ||
| 1 | ||
| 0 | ||
| 0 |
*9.3.B1 |
(a)
| (b) −3i
| (c) (−3−2i)
|
| −3+5i | ||
| 5i |
| −2−2i | ||
| −4−3i |
| −1+7i | ||
| 4+8i |
| −5−5i | ||
| −5+4i |
| −15+15i | ||
| 12+15i |
| 3+4i | ||
| −1−4i | ||
| −4+i |
| −1−18i | ||
| −5+14i | ||
| 14+5i |
*9.3.B1 |
(a)
| (b) 3i
| (c) (2+4i)
|
| 3−i | ||
| 2−4i |
| −1+i | ||
| 4−2i |
| 2 | ||
| 6−6i |
| 2−5i | ||
| 1+2i |
| 15+6i | ||
| −6+3i |
| 4−i | ||
| −3−3i | ||
| −3+2i |
| 12+14i | ||
| 6−18i | ||
| −14−8i |
*9.3.B1 |
(a)
| (b) 4i
| (c) (−1+4i)
|
| −3i | ||
| 1+5i |
| 4+5i | ||
| 3+4i |
| −4−8i | ||
| −2+i |
| −3+i | ||
| −2+4i |
| −4−12i | ||
| −16−8i |
| −5i | ||
| 2−5i | ||
| −1−3i |
| 20+5i | ||
| 18+13i | ||
| 13−i |
*9.3.B3 |
| 0 | ||
| −2i | ||
| 0 |
| 5 | ||
| 0 | ||
| −2 |
| 0 | ||
| −2 | ||
| −i |
| 3i | ||
| 3i | ||
| 3i |
| 0 | ||
| 0 | ||
| i |
| −3 | ||
| −i | ||
| 1 |
*9.3.B3 |
| 5 | ||
| 1 | ||
| −2 |
| 0 | ||
| −2 | ||
| −2i |
| 1 | ||
| −2i | ||
| 0 |
| i | ||
| i | ||
| i |
| −1 | ||
| −1 | ||
| i |
| −2 | ||
| −2i | ||
| −1 |
*9.3.B3 |
| 3 | ||
| 2 | ||
| 0 |
| 2 | ||
| −1 | ||
| 2i |
| 0 | ||
| i | ||
| 1 |
| 0 | ||
| i | ||
| 1 |
| 0 | ||
| −2 | ||
| i |
| 2i | ||
| 2i | ||
| 2i |
*9.3.B4 |
| 2 | 2 | 2i | ||
| 2i | 2i | 2 | ||
| 1+i | 1+i | −1−i |
| 0 | i | 2−2i | ||
| 1+7i | 2i | 5 | ||
| 3 | 0 | 3i |
| 1 | ||
| 0 | ||
| 0 |
| 0 | ||
| 1 | ||
| 0 |
| 0 | ||
| 0 | ||
| 1 |
| 1 | ||
| 1 | ||
| 0 |
| 0 | ||
| 0 | ||
| 1 |
| 1 | ||
| 0 | ||
| 0 |
| 0 | ||
| 1 | ||
| 1 |
| 1 | ||
| 2 | ||
| i |
| 2i | ||
| 2 | ||
| −1−i |
| 2 | ||
| 2i | ||
| 1+i |
| 2 | ||
| 2i | ||
| 1+i |
| 2i | ||
| 2 | ||
| −1−i |
| 2 | ||
| 2i | ||
| 1+i |
| 2i | ||
| 2 | ||
| −1−i |
| 2 | ||
| 2i | ||
| 1+i |
| 0 | ||
| −1 | ||
| 1 |
| −1 | ||
| 1 | ||
| 0 |
| −2 | ||
| 1 | ||
| 0 |
| −i | ||
| 0 | ||
| 1 |
The basis is the empty set. | |
| 1 | ||
| 0 | ||
| 0 |
| 0 | ||
| 1 | ||
| 0 |
| 0 | ||
| 0 | ||
| 1 |
| 1 | ||
| 11 | ||
| 11i |
| 1 | ||
| 1 | ||
| 0 |
| 0 | ||
| 0 | ||
| 1 |
| 1 | ||
| 0 | ||
| 0 |
| 0 | ||
| 1 | ||
| 5 |
| 0 | ||
| 1+7i | ||
| 3 |
| 0 | ||
| 1+7i | ||
| 3 |
| i | ||
| 2i | ||
| 0 |
| 2−2i | ||
| 5 | ||
| 3i |
| 0 | ||
| 1+7i | ||
| 3 |
| 2−2i | ||
| 5 | ||
| 3i |
| 0 | ||
| 1+7i | ||
| 3 |
| i | ||
| 2i | ||
| 0 |
| −11 | ||
| 1 | ||
| 0 |
| −11i | ||
| 0 | ||
| 1 |
| −1 | ||
| 1 | ||
| 0 |
| 0 | ||
| −1 | ||
| 5 |
The basis is the empty set. | |
*9.3.B4 |
| 4 | 4 | 4i | ||
| 4i | 4i | 4 | ||
| 2+2i | 2+2i | −2−2i |
| 0 | 2i | 4−3i | ||
| 2+6i | i | 3 | ||
| 2 | 0 | 3i |
| 1 | ||
| 4 | ||
| 2i |
| 1 | ||
| 0 | ||
| 0 |
| 0 | ||
| 1 | ||
| 2 |
| 1 | ||
| 1 | ||
| 0 |
| 0 | ||
| 0 | ||
| 1 |
| 1 | ||
| 0 | ||
| 0 |
| 0 | ||
| 1 | ||
| 0 |
| 0 | ||
| 0 | ||
| 1 |
| 4 | ||
| 4i | ||
| 2+2i |
| 4 | ||
| 4i | ||
| 2+2i |
| 4i | ||
| 4 | ||
| −2−2i |
| 4 | ||
| 4i | ||
| 2+2i |
| 4 | ||
| 4i | ||
| 2+2i |
| 4i | ||
| 4 | ||
| −2−2i |
| 4i | ||
| 4 | ||
| −2−2i |
| −1 | ||
| 1 | ||
| 0 |
| −4 | ||
| 1 | ||
| 0 |
| −2i | ||
| 0 | ||
| 1 |
| 0 | ||
| −2 | ||
| 1 |
The basis is the empty set. | |
| 1 | ||
| 2 | ||
| 0 |
| 0 | ||
| 0 | ||
| 1 |
| 1 | ||
| 11 | ||
| 9i |
| 1 | ||
| 0 | ||
| 0 |
| 0 | ||
| 1 | ||
| 0 |
| 0 | ||
| 0 | ||
| 1 |
| 1 | ||
| 0 | ||
| 0 |
| 0 | ||
| 1 | ||
| 3 |
| 0 | ||
| 2+6i | ||
| 2 |
| 0 | ||
| 2+6i | ||
| 2 |
| 2i | ||
| i | ||
| 0 |
| 4−3i | ||
| 3 | ||
| 3i |
| 0 | ||
| 2+6i | ||
| 2 |
| 2i | ||
| i | ||
| 0 |
| 0 | ||
| 2+6i | ||
| 2 |
| 4−3i | ||
| 3 | ||
| 3i |
| −1 | ||
| 2 | ||
| 0 |
| 0 | ||
| −2 | ||
| 3 |
| −11 | ||
| 1 | ||
| 0 |
| −9i | ||
| 0 | ||
| 1 |
The basis is the empty set. | |
*9.3.B4 |
| 2 | 2 | 2i | ||
| 2i | 2i | 2 | ||
| 1+i | 1+i | −1−i |
| 0 | 2i | 3−3i | ||
| 3+3i | 3i | 3 | ||
| 2 | 0 | 3i |
| 1 | ||
| 0 | ||
| 0 |
| 0 | ||
| 1 | ||
| 1 |
| 1 | ||
| 2 | ||
| i |
| 1 | ||
| 0 | ||
| 0 |
| 0 | ||
| 1 | ||
| 0 |
| 0 | ||
| 0 | ||
| 1 |
| 1 | ||
| 1 | ||
| 0 |
| 0 | ||
| 0 | ||
| 1 |
| 2 | ||
| 2i | ||
| 1+i |
| 2i | ||
| 2 | ||
| −1−i |
| 2i | ||
| 2 | ||
| −1−i |
| 2 | ||
| 2i | ||
| 1+i |
| 2 | ||
| 2i | ||
| 1+i |
| 2 | ||
| 2i | ||
| 1+i |
| 2i | ||
| 2 | ||
| −1−i |
| −1 | ||
| 1 | ||
| 0 |
| −2 | ||
| 1 | ||
| 0 |
| −i | ||
| 0 | ||
| 1 |
| 0 | ||
| −1 | ||
| 1 |
The basis is the empty set. | |
| 1 | ||
| 11 | ||
| 8i |
| 1 | ||
| 0 | ||
| 0 |
| 0 | ||
| 1 | ||
| 0 |
| 0 | ||
| 0 | ||
| 1 |
| 1 | ||
| 2 | ||
| 0 |
| 0 | ||
| 0 | ||
| 1 |
| 1 | ||
| 0 | ||
| 0 |
| 0 | ||
| 1 | ||
| 3 |
| 0 | ||
| 3+3i | ||
| 2 |
| 2i | ||
| 3i | ||
| 0 |
| 0 | ||
| 3+3i | ||
| 2 |
| 2i | ||
| 3i | ||
| 0 |
| 3−3i | ||
| 3 | ||
| 3i |
| 0 | ||
| 3+3i | ||
| 2 |
| 3−3i | ||
| 3 | ||
| 3i |
| 0 | ||
| 3+3i | ||
| 2 |
| 0 | ||
| −1 | ||
| 3 |
| −11 | ||
| 1 | ||
| 0 |
| −8i | ||
| 0 | ||
| 1 |
| −1 | ||
| 2 | ||
| 0 |
The basis is the empty set. | |